College Algebra Module 3 Review
What is the relative extrema and how can you find its ordered pairs?
A relative extrema is the min and max of a graph. The high point on a graph represents the relative maximum. Find the coordinates by looking at it on a graph. It should be an ordered pair. The low point on a graph represents the relative minimum. Find the coordinates by looking at it on a graph. It should be an ordered pair. When writing local or relative max/min, it should look like this: local min = y-value of local min. X = x-value of local min.
What is the long-run behavior of a negative, even polynomial?
As x → oo, f(x) →-oo As x →-oo, f(x) →-oo
Details of an inverse function
Inverse undo one another. Like, x - 3 has an inverse of x + 3. Invertible means a function that has an inverse. Inverse functions often look like this: f^-1 If given a problem that asks to turn something into its inverse with ordered pairs, simply change the order of the x's and y's. Ex: f(x) = {(8, 9), (3, 5)} →f(x)^-1 = {(9, 8), (5, 3)} Likewise, the domain of the original function is the range of the inverse function and vice versa. Remember that all original functions and inverse functions touch/symmetrical at the line y = x.
Methods of writing an inverse
Write it in vertex form(a(x - h)^2 + k) Make it equal y (easy method for small problems) Use the quadratic formula IF it is in vertex form or in a form where a, b, and c are all easy to find. (-b + sqrt(-b^2 - 4ac)/2, where the end product should be something like this: -b + sqrt(number +/- y)
Real numbers
a + 0i
Complex numbers
a +/- bi
What is the long-run behavior of a polynomial with an odd degree?
x→oo, f(x) →-oo x →-oo, f(x) →oo
What are the rules of the ordered pairs of a vertex form?
you must negate the sign of the h. If the polynomial in vertex form has a positive number for h, it must be negative because the formula for the vertex form is f(x) = a(x - h)^2 + k. However, the k stays the same of how it looks in the polynomial. Ex: (-h, k)
Imaginary numbers
0 + bi
How do you use Cauchy's Bond if asked?
1. First, find the largest, absolute value number in the polynomial and label it as M. 2. Now, find the absolute value of the leading term's coefficient. 3. Finally, insert them into this formula: [-M/|leading term| - 1, M/|leading term| + 1] 4. In the end, it should look like an ordered pair in brackets. The numbers should be the same, but the one on the left is negative and the other is positive.
What are the forms a polynomial can take?
1. Standard form: f(x) = ax^2 + bx + c 2. Transformation/vertex form: f(x) = a(x - h)^2 + k 3. Vertex: (h, k).
What is the long-run behavior of an odd polynomial?
As x →oo, f(x) → -oo As x →-oo, f(x) →oo
What is the long-run behavior of a negative, odd polynomial?
As x →oo, f(x) →-oo As x → -oo, f(x) → oo
What is the long-run behavior of an even polynomial?
As x →oo, f(x) →oo As x →-oo, f(x) →oo
Why are the square and cube root functions power functions?
Because they have fractional powers. Sqrt(x) = x^1/2 and cbrt(x) = x^1/3
Why are the reciprocal functions power functions?
Because they have negative whole powers. f(x) = 1/x is equivalent to x^-1, while 1/x^2 is equivalent to x^-2.
Why are the quadratic and cubic functions power functions?
Because they have whole number powers
How do you subtract complex numbers?
Before grouping the like terms, make sure to distribute -1 into the parentheses with the minus sign in front of it. Then, keep the signs and group together the like-terms to subtract. (a + bi) - (c + di) = (a - c) + (b - d)i
How do you find the least possible degree?
Count the number of spots where the graph could have been/intersect the x-axis.
What reveals the vertical intercept of a graph?
The constant term in a polynomial
What is one method to find the horizontal intercepts of a trinomial?
The quadratic formula: -b+/-sqrt(-b^2 - 4ac)/2a
How do you multiply complex numbers?
Treat it like multiplying polynomials and remember that i^2 = -1 (a + bi)(c + di) = a(c + di) + bi(c + di)
Graphically, what do alterations of the value of a in a polynomial look like?
When the value of a is changed on a graph, the parabola will either expand outward or narrow inward. If it is increased, it will narrow. If it is decreased, it will widen. Negative a values result in parabolas opening downward.
What is a horizontal asymptote?
a horizontal line y = b where the graph approaches the line as the inputs get large. As x →+/- oo, f(x) →b
How to write the polynomial formula with horizontal intercepts
f(x) = a(x - h)(x - h)(x - h) where h stands for the horizontal intercept and where a stands for the value stretched.
Rules about the finding the Domain and Range of Radical Functions
f(x) = sqrt(x) with an index of n If n is even x > 0 [x is non-negative] f(x) > 0 If n is odd x can be any real number f(x) can be any real number
What is the horizontal asymptote if the degree of the denominator and the numerator are both the same?
leading coefficient of numerator/leading coefficient of denominator
When is a vertex a minimum and when is a vertex a maximum?
A Vertex is a Maximum if the graph opens down, meaning the highest point on the parabola is when it arches high (a must be greater than 0). A vertex is a minimum if the graph opens up, meaning the lowest point is when it droops down before ascending back upward (a must be less than 0).
What is a complex number?
A complex number is the sum of a real number and an imaginary number. A complex number is a number z = a + bi, where a and b are real numbers a is the real part of the complex number b is the imaginary part of the complex number i = sqrt(-1)
How can you tell which functions can have inverses and which cannot?
A function only has an inverse if it is a one-to-one, which means each y-value is paired with only one x-value. The horizontal line test helps verify which tests are one-to-one. If a horizontal line does NOT intersect the graph of a function in more than one point, then it is not one-to-one. Make sure to test the vertical line first to ensure it is a function.
How to find out the relative extrema:
A function/graph increases if as x increases, y increases. A function/graph decreases if x increases, y decreases. An easy way to remember is to trace the graph. If you are moving up hill, the function is increasing. If you are moving downhill, the graph is decreasing.
How do you identify the leading term, the degree, and the leading coefficient of a polynomial?
A leading term is the term with the highest exponent. The degree is the highest exponent. The leading coefficient is the number in front of the leading term.
What is the Fundamental Theorem of Algebra?
A non-constant polynomial f with real or complex coefficients will have at least one real or complex zero
How does the degree of a polynomial relate to the horizontal intercepts/how to find the maximum number of real zeros?
A polynomial with a degree of n(any number) will have at most n horizontal intercepts. Whatever the degree is, that is the maximum of x-intercepts it has. Odd degree polynomials always have at least one x-intercept.
How can you determine a polynomial's multiplicity by looking at a graph alone?
If a function crosses the x-axis, then the root or zero has an odd multiplicity. If it touches the x-axis but does not cross it, then it has an even multiplicity.
What is the Complex Factorization Theorem?
If f is a polynomial f with real or complex coefficients with degree n > 1, then f has exactly n real or complex zeros, counting multiplicities.
What is a vertical asymptote?
a vertical line x = a where the graph tends towards positive or negative infinity as the inputs approach a. As x →a, f(x) →+/-oo. At points where the denominator of a rational function equals zero and the numerator is not zero, the rational function has a vertical asymptote. (x = number)
How can holes be created in graphs of rational functions?
At points where both the numerator and denominator of a rational function equal zero, factor the numerator and denominator and simplify: either the simplified function will have a vertical asymptote (and the original function will as well) or the graph of the original function will have a hole. Ex: (x - 5)(x + 4)/(x+7)(x-5). There is a hole in (x -5) or x = 5
Definition and details of rational functions
Definition: functions involving roots are radical functions. The previous example illustrated two important things: 1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one. 2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
How do you find the vertex of a quadratic if you are given a polynomial in standard form?
First, use the axis of symmetry formula(-b/2a) to get h. Secondly, insert the value of h into the original polynomial and solve. This will be your (h, k).
How do you determine the long-run behavior of a function/polynomial?
For any polynomial, the long run behavior of the polynomial will match the long run behavior of the leading term.
How do the horizontal intercept and the degree of a polynomial impact one another graphically?
For higher even powers 4, 6, 8... the graph will still bounce off the horizontal axis but the graph will appear flatter with each increasing even power as it approaches and leaves the axis. For higher odd powers 5, 7, 9... the graph will still pass through the horizontal axis but the graph will appear flatter with each increasing odd power as it approaches and leaves the axis.
How do you add complex numbers?
Group the like-terms and add: (a + bi) + (c + di) = (a + c) + (b + d)i
What do these asymptotes signify if one was trying to create a graph from these numbers?
If one was trying to create a graph from a function alone, they would need the x-intercepts, the vertical asymptote, and the horizontal asymptote. Because the vertical asymptote = the denominator's zeros, the vertical lines would be /(x - vertical line)(x - vertical line). As for the x-intercepts, these create the numerator. Their values would be plugged into the numerator like this: (x - x-intercept)(x -- x-intercept)/(x - vertical line)(x - vertical line) Finally, the horizontal asymptote would tell us what the whole fraction is being multiplied by. If the denominator has a greater degree than the numerator, then it's simply one and we do not need to write this. If the horizontal asymptote is zero, in the case that the numerator's degree is greater, then we don't need to write anything either. If the degrees of the denominator and the numerator are the same, then have that value multiplying into the fraction. (if the asymptote turns out to be a fraction, then multiply the numerator by the numerator and the denominator by the denominator)
Graphically, what do alterations of the value of h in a polynomial look like?
If the value of h is changed, it will move horizontally. Think of h like x. If it is increased, the parabola will move right. If it is decreased, it will move left. In the equation, when h is increasing, it will appear as negative. This is because of the rule addressed above. If h is decreasing or moving left, it will appear as positive in the equation.
What is the classification of all numbers?
Real Numbers: Natural numbers: 1, 12, 7, 10 Whole numbers: 3, 0, 4, 11 Integers: -6, 8, 9, -20 Rational Numbers: 3.89, 1/3, 0.42(repeating) Irrational Numbers: sqrt(2), sqrt(5), pi Complex Numbers: 3 - 4i, 7 + 2i, -isqrt(4), Sqrt(3) - 5i
How can you find the vertical asymptote?
Set the denominator to zero and solve.
What is an oblique asymptote and how is it found?
Slant asymptotes are just like normal asymptotes, but they are diagonal lines instead of being vertical or horizontal. A graph has a slant/or oblique asymptote if the degree of the numerator is one degree higher than the degree of the denominator. To find slant asymptotes, divide the numerator by the denominator and keep only the quotient. This basically means we can disregard any remainders. Draw a slant asymptote just like you would draw a slope line/normal line. It can always be found through synthetic or long division of the denominator being on the outside and the numerator being on the inside.
How is the amount of times needed to complete synthetic division determined?
The amount of times you need to repeat the process is decided by the multiplicity of a found number, such as (x - 2)^2 on a graph, then do synthetic division twice to find (x^2 + x + number)
What is a complex conjugate?
The conjugate of a complex number a + bi is the number a - bi. It is commonly used for division problems.
How can you find the maximum number of turns without drawing a graph?
The degree of the polynomial - 1 = the amount of turning points
What is the vertical intercept and how do you find it?
The vertical intercept is the y-intercept, which means it is found when the values of x = 0.
What exponent do normal numbers without variables have?
Their degree, or exponent, is zero.
What is the horizontal asymptote when the degree of the denominator is less than the degree of the numerator?
There would not be a horizontal asmyptote.
Graphically, what determines the multiplicity?
When a graph's x-intercept only touches the x-axis without crossing it, that x-intercept has an even multiplicity(exponent) When a graph's x-intercept crosses the x-axis, the x-intercept has an odd multiplicity(exponent)
Verifications of the Polynomial
When asked for a polynomial to verify your solutions, simply write your given polynomial or the polynomial you've made like this: f(x) = polynomial When asked to find verifications for your solutions, the answer will be your x-intercepts. When asked about the x-intercepts in your polynomial when your equation is written like this: f(x) = a(x - n)(x - n)(x - n), then you can easily identify them as the n's. If you want to write them as ordered pairs, remember that their signs were swapped upon entering this form. Write them as (n, 0) When asked about the zeros of the polynomial, the answer will be the same. They are the x-intercepts, but they are written as x = number and with their multiplicities
How can you complete complex division if the denominator is just an i?
When you have number/number(i), think of the denominator as number(i) + 0 to form the conjugate. This will essentially create a negated form of the number(i), which will be multiplied on top and bottom. Once you have done this, solve like an ordinary division problem.
How can you find the horizontal asmyptote?
refer to the reference above, about denominator's degree > numerator's degree = 0, numerator degree > denominator degree = no horizontal intercept, and denominator degree = numerator degree --> coefficient/coefficient
How to find the Real and Nonreal Zeros of a Polynomial
1. Begin by defining your constant and your leading coefficient. List their factors. 2. Use synthetic division to test which shared factor ends with a remainder of zero. 3. This process may be repeated multiple times depending on the multiplicity of your polynomial. For example, if you have a polynomial with a multiplicity of 5, this process will be repeated 3 times. You can stop once you have (x^2 + x + number). Make sure to write down all of your successful numbers in synthetic division in the form of (x - n), where n stands for number 4. When you have (x - n)(x^2 + x + number), you will likely be at the point where you have reached all real zeros. Before this, make sure that the three-termed parenthesis is factored and simplified to the fullest. 5. To proceed past (x^2 + x + number), you will likely be looking for nonreal zeros. With these, use the quadratic formula and solve. 6. Once you've reached your final steps, remember that you can use i(sqrt(-1)) to factor out of certain numbers. 7. When finished with finding all zeros, real and nonreal, your answer should look something like this: f(x) = (x - n)(x - n)(x - n), where all of the zeros are clearly portrayed.
How to find the long-run behavior
1. Calculate everything you need first. Whatever in the polynomial has the highest power is the leading term. This is what determines the long-run behavior. 2. Refer to 3.01, where it lists the even, even negative, odd, and odd negative long-run behavior. Depending on your work from the first step, this will determine how it moves. 3. Once you've determined how it moves in relation to the x and f(x) values, either graph or you're finished.
How to solve a polynomial inequality
1. Determine which x-values make the product equal to zero. This is done by simply negating the numbers in the parentheses or going through the process of x + number = 0 --> x = -number. Negative numbers will become positive and positive numbers will become negative. 2. Mark these points on a number line. Because these points are not less than zero, make sure the points are open and not closed. If the sign was less than or equal to, the given values would apply(closed circles). But, in this case, keep the circles open. 3. Now, we pick test values for each interval. (one for each number that you found out in step 1.) These test values are picked to satisfy the inequality. If the test values satisfy the inequality, then the interval is a part of the solution and if it does not, then the interval is not a part of the solution. To pick test values, just pick what would qualify for the specific points. 4. Create another number line and mark the points where the test values succeeded and where they did not. Draw lines to make this more obvious. 5. Write the solution using interval notation and inequalities. All of this depends on how the answer is asked to be presented. 6. Verify your answers on the graph. To find when the function is less than zero, or when y is less than zero, the y-values must be negative. Graphically, this is when the graph is below the x-axis. Essentially, this is when the graph is in the negative values of y. To help see the locations where it is below the x-axis, trace these areas on the graph. Additionally, verify your x-intercepts by spotting their locations and comparing the answers and intervals.
How do you find the vertex of a standard form polynomial using the completing the square method? (With example: 2x^2 - 12x + 14)
1. Factor out the leading coefficient: We start by factoring the leading coefficient from the quadratic and linear terms. 2(x^2 - 6x + 7) 2. Turn the parentheses into a perfect square by adding a variable: Next, we are going to add something inside the parentheses so that the quadratic inside the parentheses becomes a perfect square. In other words, we are looking for values p and q so that (x^2 - 6x + p) = (x - q)^2 3. Determine values of the new variables by splitting the middle term of the left parentheses for it to equal the variable on the right. Square the given, half of the middle coefficient on the left to get the variable in the left parentheses. Notice that if multiplied out on the right, the middle term would be -2q, so q must be half of the middle term on the left; q = -3. In that case, p must be (-3)^2 = 9. (x^2 - 6x + 9) = (x - 3)^2 4. Cancel out the newly determined values of the new variables by determining what they would equal in the parentheses and negating the given sign outside the parentheses. Now, we can't just add 9 into the expression - that would change the value of the expression. In fact, adding 9 inside the parentheses actually adds 18 to the expression, since the 2 outside the parentheses will distribute. To keep the expression balanced, we can subtract 18. 2(x^2 - 6x +9) + 14 - 18 5. Simplify and identify the vertices: Simplifying, we are left with vertex form. 2(x - 3)^2 - 4
How do you graph a rational function?
1. Find the domain of the given function and create a list of values. The domain's processes can be seen below 2. Factor both the numerator and the denominator in simplest terms. It should look like this: (x +/- number)(x +/-number)/(x +/- number)(x +/- number) when done with factoring. 3. For the domain, only focus on the denominator. Make the denominator equal zero. Make sure to take each parentheses individually. 4. Once you have your answers, write them as instructed. It helps to draw a number line to understand that your results are the only numbers the domain cannot be. Ex: answers are -2 and On a number line, plot an open circle on -2 and 4. Shade the areas behind, in between, and in front of these numbers. As an interval, it would be: (-oo, -2)U(-2, 4)U(4, oo) 5. Now that you have a domain, you can begin dealing with your function and finding asymptotes. See if there are any holes. If there are not, proceed. If there are, remove them from the function and make note of their existence. 6. To find the vertical asymptote, look to the denominator alone. Make it equal zero. This x = number will be your vertical asymptote, or a vertical line which the graphs get close to, but do not overlap. 7. Depending on what you are asked, you may need to find the horizontal asymptote. This can be done by referring to the chart. Sometimes, there are no horizontal asymptotes depending on the function. 8. If you can skip the horizontal asymptote, try to find points close to the vertical asymptote to draw your line. Two or more are necessary. It is wise to find the y and x-intercepts, but you do not have to unless asked. 9. To find the points, insert the values into the function and solve. This can also be done through the graphing calculator. 10. Draw your graph once you have points plotted and both of your asymptotes. If you have a hole, simply dot it with an open circle and continue sketching your graph
How to Find the Zeros of a Non-Factored, Large Polynomial
1. First, check if it can be factored using grouping, where the 4 term polynomial is separated into two groups of two. If it cannot be factored this way, proceed to the next step. 2. Pick the leading term's coefficient and the constant of the polynomial. Arrange them like this: + constant/coefficient. 3. Once you have these, identify all of the factors of the constant's number and the coefficient's number. Write these off to the side in a bank which will be tested individually. Remember: they are all + 4. An optional method of finding out how many zeros there are is the Carts of Rules Signs. Using this method, we bring down the signs of each term and draw lines to connect them. The amount of lines that connect can help determine the degree, or the amount of parentheses there will be. If there are 3 lines, for example, then the degree will be odd, so 1 or 3. 5. Then, test if there are negative, possible zeros by substituting -x into the polynomial function. If the result is all negative, then none of the zeros will be negative. This helps narrow down the factors. We do not have to use synthetic division on any factor that is negative. 6. Now, use synthetic division with the pool of factors from the leading term's coefficient and the constant's factors. It is a long process of elimination. If it ends with a remainder, it is not correct. If it ends without a remainder, it is correct. 7. Once you have a successful synthetic division with a remainder of zero, then set the new polynomial equal to zero. If possible, factor. If you can still not factor, then proceed to the next step. 8. If the new polynomial created by synthetic division is not factorable, use the quadratic formula(-b + sqrt(-b^2 - 4ac)/2a). This will give you two zeros. If the earlier processes suggest that there is only one zero, then you do not have to do this. But, if there is a chance of three, like in this example, then use the quadratic formula to see if there are two more zeros. 9. If the quadratic formula produces irrational/unreal numbers, (i), then do not include them because you are tasked to find real zeros.
What are the steps for long division?
1. First, identify the binomial or trinomial that is being divided into the polynomial. Then, write it out like an ordinary long division problem 2. Now, find the first term of the binomial and the polynomial. Divide the first polynomial's term by the first binomial's term. Whatever this equals must be written on the top of the long division. 3. Multiply the quotient into the binomial that is dividing into the polynomial. Once you have this, proceed to the next step. 4. Subtract the product of the quotient's multiplication into the binomial from the polynomial. Make sure to align like terms. The subtraction must be like this: -(quotient's multiplication), 5. With this answer, we repeat the second step. Locate the first term in the polynomial that has just been subtracted. Divide this by the first term in the binomial 6. Whatever the quotient is from the division, write it in the second spot, right next to the first, on top of the division bar. Now, multiply your quotient into the binomial. 7. Place the product of the quotient's multiplication beneath the division bar and prepare to subtract. Try your best to align like-terms. Subtract like this: -(quotient's multiplication). 8. You should be on your last step. If not, it is okay. Continue this process. Once you have three terms on the division bar, you will likely be left with a single number without a variable. This is your remainder.
How do you write a polynomial by simply looking at a graph?
1. First, observe the graph. Look for x-intercepts, y-intercepts, and its direction. Is it pointed up or down? Has it moved from the origin? 2. After finding out the important information, we choose the type of form the function will be. Sometimes, the question will give this. In any sense, it is best to find the vertex form(a(x - h)^2 +k) from a graph 3. Calculate however many spaces it moved horizontally and plug it into the h value. (make sure to watch the signs) 4. Calculcate however many spaces it moved vertically. and plug it into the k value 5. Now, you will likely have a(x -/+ number)^2 + (number). Find out the value of a by using points on the graph. Ideally, use the y-intercept if you have it. 6. Set the y-intercept to its value and set all of x to 0. Solve and simplify to find out the value of a. 7. If you need the polynomial to be in standard form, simply multiply and simplify. You will have ax^2 + bx + c in the end.
How to find an even root:
1. First, observe the index, or whatever type of root the problems have. To illustrate, my example has a square root of 2, a cube root of 3, and a root of 4. Determine whether the index is even or odd 2. After observing the rules, mark which ones are even and which ones are odd. 3. Once you've clarified this, write it accordingly. If you have a radicand(number beneath the root) with an even root, then input it in the given formula: x > 0. 4. Solve after inputting. This should be a simple use of algebra that will give you the domain. Remember: dividing by a negative flips the inequality. 5. For finding the range, the process is easier. If your example is a non-negative number, then your range will simply be what it states in the rules of even: f(x) > 0. 6. To verify the domain and range, you can always graph the function.
How to find an odd root:
1. First, observe the index, or whatever type of root the problems have. To illustrate, my example has a square root of 2, a cube root of 3, and a root of 4. Determine whether the index is even or odd 2. After observing the rules, mark which ones are even and which ones are odd. 3. Unlike with even roots, there is no formula for the function to be inserted in. Rather, understand that the x can be any real number, which means the answer for the domain is any real number. (interval notation for all reals: (-oo, oo)) 4. The same applies for the range. The odd index means the f(x) can be all reals as well without having to do any work prior.
How to solve fraction inverse functions(x+/-number)/(x+/-number)
1. First, switch your variables. Where x is, put y. Make it so that it looks like x = (y +/- number)/y +/-number) 2. Now that it is properly arranged, multiply the denominator on both sides. This will look like denominator/1 * x = (y +/- number)/(y +/- number) * denominator/1 3. On the side with the y, it will simply cancel out, leaving you with the numerator alone. This is because of cross multiplication. 4. For the x, distribute it into the denominator. 5. Once distributed, move the y variable on the other side of the equal sign to the side with the x-values, leaving only a constant on the other side. 6. Once you've done this, factor out the y's from the x side. It should look something like this: y(numberx +/- number) = number. 7. Divide the parentheses from the number on the other side of the equal sign where it looks like this: y = number/numberx +/- number. 8. Now, you have your answer. This is the inverse. To check, you can always multiply the original by your answer and see if they cancel each other out.
How can you write a polynomial with only the intercepts? Factored form(a quadratic function with zeros r1 and r2): f(x) = a(x - r1)(x - r2) General/standard form: ax^2 + bx + c Vertex/transformation form: a(x - h)^2 + k
1. In this case, we will have three points in total: two are x-intercepts and one is the y-intercept. With these points, we will find the general/standard form of a polynomial using the factored form shown above. 2. Write out the zeros of the function. These will be what the numbers in the x-intercepts are. 3. With the x-intercept zeros identified, substitute them into the factored form for r1 and r2. Make sure to remember that both of these r variables are negative. Watch your signs. 4. Use the y-intercept to determine the value of a. Using the ordered pair of the y-intercept, understand that f(x) and y are the same. If the x of y is zero, then it would be the f(0). Whatever that value y is when x is zero, substitute f(0) for f(x) in the factored form. This will also result in the x's in the parentheses being zero. 5. With zeros substituted in the factored form and f(0) equaling the value of y, solve for a. The factored form should look like this: f(0) = a(0 - x-intercept)(0 - x-intercept) → y-intercept = a (x-intercept)(x-intercept). 6. With all of the variables known, input the information in factored form. If it asks for factored form, then your task is finished. If it asks for standard form, move to the next step. 7. Multiply the factored form until you have the general/standard form shown above.
How can you simplify a polynomial when you cannot factor ordinarily?
1. Look at the equation itself and notice the numbers on the inside and outside. 2. Now, look at the interior of the parentheses alone. Look at the central value and the last value. What numbers make the value in the middle? It has to be an addition or subtraction of one or two numbers. Once you have determined what creates the middle number, keep this in your head or write it to the side. 3. Now, do the same with the last number. What numbers create it by multiplying? These should be the same number(s) as the one that created the central term. If these are the same, then you are correct. 4. Write out your parentheses like this: (x -/+ number)(x +/- number). If the number is the same in terms of the sign and the number itself, then just write (x -/+ number)^2. 5. You can check the answer by solving it and seeing if you get the polynomial you started with. This is helpful with multiple choice questions.
How to find the least possible degree of a polynomial using a graph (If a function has degree n, the function has at most n x-intercepts(horizontal intercepts) and it has at most (n - 1) turns.)
1. Observe your graph. How many x-intercepts does it have? 2. How many turns does it have(how many times does it change from decreasing to increasing or vice versa? 3. Remember that a degree one function has only one x-intercept it would be a line. A degree two function would be a parabola, which would typically mean two x-intercepts. When trying to find the least possible degree, trust the turns instead of the x-intercepts. And treat the (n - 1) like the n is negative or the 1 is positive. If there are two turns, the least possible degree would be three (-2 - 1) = (-3).
How do you use synthetic division?
1. When you set it up, make k positive. Even though it is negative, it must be positive when it is set up. 2. Bring the a value down 3. Multiply that a value by k and put it beneath b. 4. Add the upper b to the product of k and a's multiplication and place it below. 5. Multiply the b + ka by k and place it beneath c. 6. Add the upper c to the product k's multiplication to b + ka and place it beneath c. 7. Finally, multiply the bottom number of c by k and place it beneath d. 8. Add d to the product of kc and bring it below. 9. All of the numbers below are the coefficients. Remember: if a had a degree of 3, then the number below was a degree of 2. d is the remainder and must be a fraction like number/binomial
How do you divide complex numbers?
1. Write the problem in fractional form 2. Rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. (remember that a complex number times its conjugate will give a real number. This process will remove i from the denominator) 3. Separate the fraction into two different fractions so that they can be properly simplified. (a+bi)/c+di) --> (a+bi)(c-di)/(c+di)(c-di)
What is the gravity formula?
H(t) = -16t^2 + st + h, where s stands for speed per second and h stands for how high it was dropped.
What is the horizontal asymptote when the degree of the denominator is greater than the degree of the numerator?
Horizontal asymptote: y = 0
Can horizontal asymptotes be touched? What about vertical asymptotes?
Horizontal asymptotes can be crossed. Vertical asymptotes cannot be. The graph will get close, but will never touch.
What are the toolkit functions that can be inversed without their domain being restricted?
Identity function f(x) = x; the cubic function f(x) = x^3; the square root function f(x) = sqrt(x); and the cube root function f(x) = cbrt(x).
How can you tell if the polynomial's vertex is a minimum or a maximum is not in vertex form(a(x - h)^2 + k)?
If given ordered pairs, then check the y-values. If the y-values are less than one, the vertex is a minimum. if the y-values are greater than one, then the vertex is a maximum.
Graphically, what do alterations of the value of k in a polynomial look like?
If the value of k is changed, it will move vertically. Think of k like y. If it is increased, the parabola will move upward. If it is decreased, it will move downward.
How to find the domain of a rational function
In order to determine the domain of a rational function, we want to find the x-values that we must exclude from the domain. Remember that the fraction's bar means division and division by zero always results in an undefined answer. Sometimes rational functions will simplify because they have common factors on the numerator and denominator; however, these should NOT be simplified. Do not simplify prior to finding range and domain. It will mess up the results. When you simplify, you are creating a vertical asymptote, which is unneeded for determining domain and range. 1. (When necessary, factor out the greatest common factor) 2. Set the denominator equal to zero 3. Once you use algebra to solve, remember that this number that was found must be excluded from the domain. However, it will be written in the parentheses or brackets. If it is written, then you can also write all reals except number. In interval notation, it would be written as (-oo, number)U(number, oo) 4. On a number line, you would place an open dot on the number received from step 2. If it can be all real numbers except this one, then shade on both sides for numbers that could be in the domain.
Which toolkit functions are power functions?
The constant functions f(x) = x^0; the identity function f(x) = x^1; the quadratic function f(x) = x^2; the cubic function f(x) = x^3; the reciprocal function f(x) = 1/x; the reciprocal squared function f(x) = 1/x^2; the square root function: f(x) = sqrt(x); and the cubic root function f(x) = cbrt(x).
How can you find the degree of a factored function? (Ex: (x + n)(x + n) where n stands for any number)
The degree of a function is how many parentheses there are. For example, if the graph/function is supposed to have a degree of 4 in the given, then it will have (x - h)(x - h)(x - h)(x - h). This could also be written as (x - h)^4, (x - h)^2 (x - h) (x - h), or (x - h)^3 (x- h). The degree and the number of parentheses MUST match up.
What are the rules of synthetic division?
The first major rule is that if you are dividing a binomial with a high degree, then you must fill in the spaces for the other degrees. For example, if there was x^4 + 9, then your syntehtic division must be x^4 + 0x^3 + 0x^2 + 0x + 9. The second major rule is that if you are dividing a number like 2x + 4 into a polynomial, you have to simplify this first. It has to be in the form (x - n), so it would be 2(x + 2). -2 would divide into the polynomial.
What is the horizontal intercept and how do you find it?
The horizontal intercept is the x-intercept. Often, there are multiple of these. They can also be called zeros. They are found by setting the y-value to zero, which can be found through synthetic division, factoring, completing the square, using the quadratic fraction, or other methods.
What is an imaginary number?
The most basic complex number is i, defined to be i = sqrt(-1) commonly called an imaginary number. Any real multiple of i is also an imaginary number
What is a vertex?
The part in the graph where the graph changes direction.
What is the Rational Roots Theorem and when is it used?
The rational roots theorem is used for finding the zeros of large polynomials that cannot be factored or grouped. 1. Identify the leading term's coefficient and the constant. 2. Make a list of their factors. Make sure the list is +/- 3. Once you have this, you will begin testing by either using synthetic division or by substituting the factors into the original polynomial. Either option works. Just remember: the number is only a zero if the remainder(synthetic division) or the answer is zero. 4. Once you have found the number that is a zero, you will either have a product from synthetic division or you will have to use this number to complete synthetic division. 5. If you already have your synthetic division's answer, remember to write it as a polynomial with all of the x's one degree lower than before. For example, if it began with degree 4, then the product of synthetic division would be numberx^3. 6. In the case where the degree is large, we repeat synthetic division until we have ax^2 + bx + c. 7. Once we have this, we can either factor it if possible through grouping or normal factoring, or use the quadratic formula. 8. By now, you should have all of your zeros. Likely, there will be three: the zero found through the rational roots theorem that was used in synthetic division and the others found through factoring or the quadratic formula.
How do you use the Remainder Theorem?
The remainder theorem inserts the factors of the constant and the leading term into the original polynomial and tests whether or not it is a zero. 1. The first step of this is to define the constant and the leading term's coefficient. Write out their factors and have it organized like this: + constant/coefficient. 2. Now, choose one of the factors to plug into the polynomial. Test it by using algebra. If it equals a number, then it is not a zero. If it equals zero, then it is a zero. 3. Now that you have found a zero, use synthetic division for your original polynomial. The remainder must be zero. This creates another polynomial which likely can be easily factored. 4. If needed, write the number that worked in synthetic division and in the remainder theorem as a factor. Factor form is this: (x - number). 5. If your polynomial created by the synthetic division of your known zero cannot be factored, then you have to use the quadratic formula or complete the square. How to complete the square is shown in the examples below. However, this may result in unreal zeros.
