Algebra 2: Manipulating and Interpreting Expressions
the cube of the difference of 5 times x and 4
(5x-4)^3
perfect cube
(a + b)3 = (a³ + 3a²b + 3ab² + b³)
perfect squares
(a + b)² = (a² + 2ab + b²)
Perfect square Identity
(a + b)³ = a³ + 3a²b + 3ab² + b³ (2x + 3y²)³: a = 2x, and b = 3y² (2x + 3y²)³ = (2x)³ + 3(2x)²(3y²) + 3(2x)(3y2)² + (3y²)³ = 8x³ + 3(4x²)(3y²) + 3(2x)(9y⁴) + 27y⁶ = 8x³+ 36x²y² + 54xy⁴ + 27y⁶
difference of squares
(a² − b²) = (a + b)(a − b)
sum of cubes
(a³ + b³) = (a + b)(a² − ab + b²)
difference of cubes
(a³ − b³) = (a − b)(a² + ab + b²)
factor
(noun) a part of a term that is being multiplied by another part or other parts of a term
Correct Ex. rational expression in simplest form
(x+1)(x+2) / (x+1)(x-3) = (x+2) / (x-3) x(x-3) / x(x+3) = x-3/x+3 5/ 5x + 20 = 5/ 5(x+4) = 1/ x+4 2x²/4x = 2*x*x /2*2*x = x/2
(x+y)²
(x+y)²= x²+2xy+y²
(x+y)²
(x³+3x²y+3xy²+y³)
Let R be a system consisting of rational expressions. Which operations are closed for R?
-addition, subtraction, multiplication, and division A set is said to be closed under an operation when the application of the operation between any two elements of the set leads to an element that belongs to the same set. If a set is closed under an operation, it is said to have the closure property of that operation. When we combine two rational expressions by adding, subtracting, multiplying, or dividing, we get a rational expression. This pattern indicates that rational expressions are closed for all four operations.
perfect squares 1. (a + b)² = (a² + 2ab + b²) 2. (a + b)² = (a + b)(a + b) = a² + ab + ab + b² = a² + 2ab + b²
1. The perfect square identity is the most frequently used polynomial identity. To prove this identity, again we'll show that the left side, (a + b)², equals the right side, (a² + 2ab + b²). 2. Multiply the terms on the left side of the identity. Add the like terms. This time we don't cancel like terms with opposite values as coefficients, but we do arrive at the right side of the identity.
perfect cube 1. (a + b)³ = (a³ + 3a²b + 3ab² + b³) 2. (a + b)³ = (a + b)(a + b)(a + b) = (a + b)(a² + ab + ab + b²) = (a + b)(a² + 2ab + b²) = a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + (2a²b + a²b) + (ab² + 2ab²) + b³ = a³ + 3a²b + 3ab² + b³
1. To prove this identity, show that the left side, (a + b)³, equals the right side, (a³ + 3a²b + 3a²2 + b³). 2. Multiply the terms on the left side of the identity using the distributive property. Simplify using the commutative property. Add like terms, and arrive at the right side of the identity.
difference of cubes 1. (a³ − b³) = (a − b)(a² + ab + b²) 2. (a − b)(a² + ab + b²) = a³ + a²b + ab² − a²b − ab² − b³ = a³ + (a²b − a²b) + (ab² − ab²) − b³ = a³ − b³
1. To prove this identity, we'll show that the right side, (a − b) (a² + ab + b²), equals the left side, (a³ − b³). 2. Multiply the factors of the right side of the identity by distributing terms. Simplify, using the commutative property and combining like terms.
sum of cubes 1. (a³ + b³) = (a + b)(a² − ab + b²) 2. (a + b)(a²− ab + b²) = a³ − a²b + ab² + a²b − ab² + b³ = a³ + (-a²b + a²b) + (ab² − ab²) + b³ = a³ + b³
1. To prove this identity, we'll show that the right side, (a + b)(a² − ab + b²), equals the left side, (a³ + b³). 2. Multiply the factors on the right side of the identity by distributing terms. Simplify using the commutative property. Cancel additive inverses.
difference of squares 1. (a² − b² ) = (a + b)(a − b) 2. (a + b)(a − b) = a(a − b) + b(a − b) = a2 − ab + ab − b2 = a2 − b2
1. We use this polynomial identity for factoring a difference of two squares. To prove this identity, we'll show that the right side, (a + b)(a − b), equals the left side, (a² − b²). 2. Multiply the terms on the right side of the equation. Distribute and simplify. Cancel out the additive inverses to get the left side of the identity.
Finding Least Common multiple of an Algebraic Expression 1/ x²+4x-12 + 1/x²-5x+6 x²+4x-12=(x+6)(x-2) x²-5x+6=(x-2)(x-3) x²+4x-12=(x+6)(x-2) x²-5x+6=(x-3) (x+6)(x-2)(x-3)
1/ x²+4x-12 + 1/x²-5x+6 1. Denominator with factors 2. Deleting repeated factors 3. LCM - The LCM of the two algebraic expressions is (x + 6)(x − 2)(x − 3).
4y(3x²-8x+2)
12x²y-32xy+8y
A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 50 percent of its previous height. The total vertical distance down the ball has traveled when it hits the ground the fifth time is meters.
19.375
Binomial
2 terms
simplifying the expression 2(x+1) / x²-x-2 + x / x²-3x+2 = 2(x+1) / (x+1)(x-2) + x/(x-1)(x-2) = 2/x-2 + x/(x-1)(x-2) = 2/x-2 x (x-1)(x-1) + x / (x-1)(x-2) =2(x-1) + x / (x-2)(x-1) = 3x-2/ (x-2)(x-1)
2(x+1) / x²-x-2 + x / x²-3x+2 1. Factor the denominators 2. Remove the common factors of each rational expression. 3. Find the LCM, and change to the common denominator.
(3y-7)(9y²+21y+49)
27y³-343
2x+6 / x²+x-6 = 2(x+3) / (x+3)(x-2) = 2(x+3) / (x+3)(x-2) = 2/ x-2
2x+6 / x² + x - 6 1. In the numerator, pull the GCF, 2, out of the parentheses. Rewrite the trinomial in the denominator as a product of its factors. 2. Cancel out the common factor, (x + 3), in the numerator and the denominator. 3. This is the simplest form of the rational expression.
Ex. polynomial
2x2 + x is a polynomial with two terms combined by addition. The first term, 2x2, has a coefficient of 2, and the second term, x, has a coefficient of 1.
2x²- 3x-2 / 2x² - x-1 = (2x+1)(x-2) / (x-1)( 2x+1) = (2x+1)(x-2) / (x-1)( 2x+1) = x-2/x-1
2x²- 3x-2 / 2x² - x-1 1. Factor the trinomials in the numerator and the denominator. 2. Cancel out the common factors. 3. This is the simplest form of rational expression.
Trinomial
3 terms
Find and pull out the greatest common factor among the terms in the expression, and then factor the trinomial: 3x³ + 3x² - 18x = 3x(x²+ x - 6) = 3x(x²+3x-2x-6) = 3x(x(x+3)-2(x+3)) = 3x(x + 3)(x - 2) 3x³ + 3x² - 18x = 3x(x + 3)(x - 2)
3x³ + 3x² - 18x
(6x − 4)3
4 less than 6x, cubed
(6x)^3 − 4
4 less than the 3rd power of the product of 6 and x
4x2 + 14x + 6 = 2(2x^2 + 7x + 3) ax2 + bx + c 2(2x^2 + 7x + 3) Factors of 6: 1 and 6, 2 and 3 2(2x^2 + 7x + 3) = 2(2x^2 + 6x + x + 3) 2(2x^2 + 7x + 3) = 2(2x^2 + 6x + x + 3) = 2[(2x^2 + 6x) + (x + 3)] 2[(2x^2 + 6x) + (x + 3)] = 2[2x(x + 3) + 1(x + 3)] 2[2x(x + 3) + 1(x + 3)] 2[2x(x + 3) + 1(x + 3)] = 2(x + 3)(2x + 1) 4x^2 + 14x + 6 is 2(x + 3)(2x + 1).
4x^2 + 14x + 6. When attempting to factor an expression where the leading coefficient isn't 1, first check whether there's a greatest common factor (GCF) that can be factored out of the entire expression. In this case, each coefficient is a multiple of 2, so factor a 2 out of each term: Next, use grouping to factor this expression. Compare the part enclosed within the parentheses with the standard form of the expression: To factor this expression by grouping, first multiply the a and c values and then use the technique for factoring basic quadratics to find the factors of this product that add to b. The product of a (2) and c (3) is 6. The factors of 6 are 1 and 6 and 2 and 3. Use the factor pair 1 and 6 because they add to 7. However, instead of just substituting these values into the factored form as we do when the value of a is 1, we rewrite the b term of the expression in terms of the factors of 6. Now, we want to view this expression as two terms. The first term is (2x2 + 6), and the second term is (x + 3). Next, look for the GCF of the terms. The GCF of the first term is 2x, and the GCF of the second term is just 1 (as there is nothing in common). Write these GCFs in front of each term. Now there's a common factor in both terms, (x + 3). To finish factoring this expression, pull out the (x + 3) term. This leaves (2x + 1) as the remaining factor. Don't forget to include the initial GCF, 2, in front of the expression.
4x²-12x-16 / 2x³-8x²+_6x-24 = 4(x²-3x-4) / 2(x³-4x²+3x-12) =4(x-4)(x+1) / 2(x²(x-4) + 3(x-4) = 4(x-4)(x+1) / 2(x²+3)(x-4) = 2(x+1) / (x²+3)
4x³-12x-16 / 2x²-8x²_6x-24 1. Pull the GCF out of the numerator (4) and the denominator (2). 2. Factor the trinomial in the numerator. Rewrite the denominator by grouping by a common factor. 3. Write the denominator as a product of factors. 4. Cancel out the common factor in the numerator and the denominator, and simplify. 5. This is the simplest form of the rational expression. All GCFs and factors have been canceled, and there is no other way to write the expression in simpler form.
example of a rational expression in a more complicated form
5/x + x-10/ 4x^3 - 6
the quotient of the difference of 5 times x cubed and 4 and x
5x^3 - 4 ----------- x
The difference of 5 times the cube of x cubed and the quotient of 4 times x and 3
5x^3 - 4x/3
5 times the cube of x divided by 4 times x
5x^3 / 4x
7(x+8)³
7 times the cube of the sum of x and 8
What is the sum of the first eight terms of a geometric series whose first term is 3 and whose common ratio is ?
765/128
(7x)³ + 8
8 added to the cube of 7x
7x³ + 8
8 added to the product of 7 and x cubed
At the beginning of every year, Molly deposits $200 in a savings account that offers an interest rate of 20%, compounded annually. The total amount that Molly will have in her account at the end of 3 years is $.
873.60
Manipulate the expression for the average cost by splitting it into two parts: 5.35x+40/x = 5.35x / x + 40/x = 5.35 + 40/x The basic package without any premium channels costs $40, and x represents the number of premium channels. So, the quotient 40/x represents the cost of the basic package per additional premium channel.
A cable service provider charges $40 per month for its basic package plus an additional $5.35 for each premium channel chosen. The average cost for cable per additional channel is given by the expression (5.35x + 40) / x, where x is the number of premium channels added to the basic package. What does the quotient 40/x represent?
It is given that that length of the container is half of a foot longer than its width and that x is the width of the container. So, the length of the container can be represented by the expression below. 1/2 + x Recall that the formula for the volume of a rectangular prism is given by the formula V = l · w · h, where l is the length, w is the width, and h is the height. Rewrite the given expression by factoring out the leading coefficient and x². The expression left inside the parentheses should have a linear term with a leading coefficient of 1 to match the expression for the length of the container. 7x³ + 3.5x² 7x²(x+1.2) Now, rewrite this resulting expression as a product of three factors to represent the width, length, and height of the container. 7x * x * (x+1/2) Thus, if the length of the container is half a foot longer than its width, x, then the expression 7x represents the height of the container.
A high school Philanthropy Club is collecting plastic waste which will be used to create benches for a local park. The volume of the rectangular container, in cubic feet, used to hold the waste is modeled by the expression below, where x is the width of the container, in feet. 7x³ + 3.5x² If the length of the container is half of a foot longer than its width, what expression represents the height of the container?
rational number
A number that can be written as a fraction
Monomial
A number, a variable, or a product of a number and one or more variables
rational expression in simplest form
A rational expression is in simplest form when the numerator and the denominator don't have any common factors. We can rewrite a rational expression so it's in simplest form by canceling out the common factors in the numerator and the denominator.
recursive formula
A recursive formula is a rule that relates each term of a sequence, after the first term, to the terms before it. In other words, depends a n depends on a n-1 ; Once the first term of a geometric sequence is defined, the recursive formula allows us to find the terms of the sequence based on the previous term and the common ratio, .
quadratic terms
A term in a quadratic equation with a variable squared
explicit formula
An explicit formula is a rule that relates each term of the sequence to the term number in the sequence. This formula allows us to find the value of any term in the sequence without needing to know any of the other terms except for the first term.
rational numbers
Any number that can be expressed as a fraction; They are real numbers in the form a/b, where a is any integer and b is a nonzero integer. Rational numbers are closed under addition, subtraction, multiplication, and division. This means adding, subtracting, multiplying, or dividing two rational numbers will result in a rational number.
standard form of a linear equation
Ax + By = C, where A,B, and C are real numbers, and A and B are not both zero.
The binomial expression 2w + 6 represents the length of the fish tank.
Dylan has a fish tank at home. He knows its height, h, is one inch less than twice the width, w, and the length, l, of the fish tank is seven inches longer than the height. Which of the following statements is true?
.ax2 + bx + cx2 + bx + c ↓ (x + p)(x + q) x2 + bx + c x2 + 9x + 14 Factors of 14: 14 and 1, 7 and 2 x2 + 9x + 14 = (x + 2)(x + 7) or x2+ 9x+ 14 = (x+ 7)(x+ 2)
Expression- x^2 + 9x + 14. To factor a quadratic expression like this one, convert it from standard form (ax^2 + bx + c), to factored form (x + p)(x + q). In this case, a is equal to 1, so look for values of p and q that are the factors of c and also add up to b. To factor this expression, look for factors of c (14), that add up to b (9). Let's consider all the factors of 14. We know we have 1 and 14 as factors. Next, we check to see whether 2 can divide evenly from 14, and get the factors 2 and 7. No other values divide evenly from 14. Therefore, the pairs of factors of 14 are 1 and 14 and 2 and 7. We're looking for factors of c we can add to get b, so we use 2 and 7, which add to 9. Now write the expression in factored form, with p and q equal to 2 and 7, respectively. Thanks to the commutative property of multiplication, we can write this expression as either (x + 2)(x + 7) or (x + 7)(x + 2). Then put the factored expression back into a standard form to make sure it was factored correctly.
x/ x+1 + 2/x = x*x / x(x+1) + 2(x+1) / x(x+1) = x*x+2(x+1) / x(x+1) =x^2 + 2x+2 / x(x+1)
Find the common denominator, , add the numerators, and simplify:
x/ x+1 - 2/x = x*x / x(x+1) 2(x+1) / x(x+1) = x*x - 2(x+1) /x(x+1) = x^2 -2x -2 / x(x+1)
Find the common denominator, , subtract the numerators, and simplify:
Rational Expressions in Different forms
Instead of just one polynomial divided by another, rational expressions can include more than one fraction, as well as additional constant terms.
infinite geometric series
It might seem impossible to determine the sum of a series that goes on for infinity. However, as long as the common ratio is between 0 and 1, it's possible to find the sum of an infinite number of terms. If |r| > 1: As n gets larger, the value of |rn| will get larger. So, the absolute value of the entire sum will get larger. In this case, the sum of the infinite series cannot be evaluated because it is infinity. If 0 < |r|< 1: As n gets larger, the value of rn gets closer to zero.
1. The average profit made from the tour (18x+35) / x, , can be found by dividing the total profit made from the tour by the number of people attending the tour. So, the expression in the numerator, 18x + 35 , represents the total profit made from the tour. 2. The denominator x represents the number of people going on the tour. 3. 18x/ x + 35/x = 18 + 35/x 4. In the fraction 35/x, the numerator,35 , represents a fixed cost, like a cover charge for the tour, that is not affected by the number of people attending the tour. So, the quotient 35/x represents the profit that the tour company makes from the cover charge per person.
Jessica is organizing a guided tour of the rain forest. The average profit per person that the touring company makes is given by the rational expression (18x + 35)/x , where x is the number of people going on the tour. 1. What does the numerator of this rational expression represent? 2. What does the denominator of this rational expression represent? 3. Rewrite the expression as a sum of two fractions, and simplify. 4. What does the quotient 35/x represent?
Joyce's average speed from the library to the gym
Joyce rode her bicycle 4 miles from her house to the library, and she rode the same distance from the library to the gym. Her average speed from the library to the gym was 15 miles per hour less than her average speed from her house to the library. Joyce modeled the trip from her house to the gym with the following expression. 4/x + 4/x-15 What does x - 15 represent in this situation?
Polynomial are not closed when dividing
Many examples can prove that they are not. Any division of polynomials that leaves a variable to the first or higher power in the denominator is not a polynomial because it has a variable with an exponent that is not a positive integer.
Let l be the length, w be the width, and d be the depth of the swimming pool. It is given that the width of the pool is twice the depth. So, the width of the pool can be represented as w = 2d. Next, it is given that the length of the pool is 3 feet longer than the width. So, the length of the pool can be represented as l = w + 3 = 2d + 3. Therefore, the binomial 2d + 3 represents the length of the swimming pool.
Mr. Wilson is building a swimming pool in his backyard. The width of the pool is twice the depth and the length of the pool is 3 feet longer than the width. Which of the following statements is true?
x/ x+1 x 2/x = x*2 / (x+1) * x = 2/ (x+1)
Multiply the numerators and the denominators, and cancel out like terms to simplify:
irrational numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.
Quartic Monomial
One term polynomial raised to the 4th power
degree 2
Quadratic
What is the simplest form of this binomial expression? a4 − b4
Rearrange the polynomial a⁴ − b⁴ in the form (a²)² − (b²)². Then use the difference of squares to factor the polynomial: a⁴ − b⁴ = (a²)² − (b²)² = (a² − b²)(a² + b²) = (a + b)(a − b)(a² + b²)
Recursive Formula for a Geometric Sequence
Relates any term in a sequence to only the prior term and the common ratio. a 1 = first term r = common ratio r(a n-1); for n is greater than or equal to 2.
8x⁶ + 27y³
Rewrite the expression so it looks like the identity. Substitute a = 2x² and b = 3y in the sum of cubes identity: 8x⁶ + 27y³= (2x²)³+ (3y)³ = (2x² + 3y)[(2x²)² − (2x²)(3y) + (3y)²] = (2x² + 3y)(4x⁴ − 6x²y+ 9y²)
1. number of group members when the site launches. 2. number of members per group after x months
Richard is designing a new website for students to create study groups and collaborate with other students. The site will launch with five study groups, each with its creator as its only member, and Richard anticipates that two study groups will be added every month. He also estimates that seven new members will be added to each study group every month. The estimate for the total number of group members is given by the polynomial expression below, where x is the number of months since the site's launch. 14x² + 37x +5 1. The constant of the polynomial expression represents the 2. The binomial (1 + 7x) is a factor of the polynomial expression and represents
1. 5/6 = 2+3/6 = 2/6 + 3/6 2. 1.25p-31/ p = 1.25p/ p - 31/p = 1.25- 31/p
Sometimes it's useful to manipulate a rational expression by splitting it into two parts. For example, in the following fraction we can work backward to go from one fraction to two fractions: 1. We can use this same process to rewrite the rational expression for the average profit per pastry from the example in the introduction. Let's split the expression in two and see what we get: 2. Notice that the constant in the new expression represents the price of each pastry. So, the expression shows that the average profit is equal to the price per pastry minus the fraction of the total cost per pastry sold.
Extended Form of Identity- perfect square identity
Substitute a² for a and b² for b: a⁴ + 2a²b² + b⁴ = (a²)2 + 2(a²)(b²)+ (b²)² = (a² + b²)² x² + 32x² + 256; let a = x² and b = 4²= 16 = (x²)² + 2(4²)(x²) + (4²)² = (x²+ 4²)² = (x²+ 16)²
constant terms
Terms in an expression that are not combined with a variable. This value does not change.
A moving company charges $115 to rent a truck for a three-day period, plus an additional $0.83 for each mile driven. The average cost per mile for a three-day period is given by the rational expression , where x is the number of miles driven during the rental period.
The company charges $115 as a rental fee and $0.83 for each mile driven. So, the expression in the numerator, 115 + 0.83x, represents the total cost for renting the truck over a three-day period.
A cab company offers a special discount on fare to senior citizens. The following expression models the average amount a cab driver of the company collects on a particular day, where x represents the number of senior citizens who travel by the company's cabs. 180x/x+4 +250 What does the constant term in the above expression represent?
The constant 250 represents the average amount a cab driver collects on a particular day when no senior citizens travel by the company's cabs.
Michael is designing a backyard garden. He would like the length of the garden to be 3 feet longer than the width. The area of Michael's garden is given by x2 + 3x, where x represents the width of the garden. The binomial x + 3 is a factor of the polynomial.
The expression x2 + 3x can be factored as x(x + 3): x2 + 3x = x(x + 3). The area of the garden is a product of length and width: area = length • width. If x represents the width, x + 3 must represent the length of the garden.
The volume of a cylindrical soap dispenser is modeled by the expression below. pie(x+1)^2(4x+1)
The factor 4x + 1 represents the height of the soap dispenser.
Factor the expression x2 + 4x − 12.
The first term is x2, so the factored form will be (x + a)(x + b) for some constants a and b. To get a middle term of 4x, the sum of a and b must be 4. To get the last term of -12 in the polynomial, the product of a and b must be 12. Therefore, the factored form is (x − 2)(x + 6).
The given expression represents the total cost of the field trip, where x is the number of students. To determine the total cost of the trip if 0 students are signed up, substitute x = 0 into the expression and evaluate. (80(0)/ 0+10) + 45 = 0/10 + 45 Thus, if the number of students signed up is zero, the school still needs to pay $45 for the field trip. Therefore, the constant 45 represents the cost of the field trip if 0 students signed up.
The following expression models the total cost of the field trip, where x represents the number of students. (80x /x+10) + 45 What does the constant term in the above expression represents?
The constant -10 represents the cost of setting up the food stall even if no cheeseburgers were sold.
The following expression models the total money earned by setting up a food stall at a funfair, where x represents the number of cheeseburgers sold. (60x/ x+3) - 10
standard form of a polynomial
The form of a polynomial that places the terms in descending order by degree.
least common denominator (LCD)
The least common multiple of the denominators of two or more fractions.
least common multiple (LCM)
The least number that is a common multiple of two or more numbers.
If we substitute y = 0 in the expression, we get this constant value: 378. Because y represents the number of years since 2010, the constant 378 represents the fox population in 2010, which is 0 years after 2010.
The population of artic foxes in a wildlife reserve since the year 2010 can be represented by the rational expression (35y/y+7) + 378, where y is the number of years since 2010. What does the constant term 378 represent?
True
The process of factoring cubic polynomials is the same as factoring quadratic polynomials. The first step is to look for the greatest common factor and then factor the rest of the expression.
factorial
The product of all whole numbers except zero that are less than or equal to a number
Closed set
The product of polynomials can have the same original variables as the factors but with higher integer exponents. These products will also be polynomials.
intergers
The set of whole numbers and their opposites
leading term
The term with the highest exponent of the polynomial
The greatest common factor of the expression πr3 + 7πr2 is πr2. So, factor out πr2 to get πr2(r + 7). Comparing this with the formula for the volume of a cylinder, we see that the height is given by (r + 7). So, the height of the cylinder is 7 units greater than its radius.
The volume of a cylindrical soup can is represented by the expression πr3 + 7πr2, where r is the radius of the cylinder. Which statement best interprets the factor (r + 7) in this context? Hint: The volume of a cylinder is given by the general formula V = πr2h, where h is the height of the cylinder.
Cubic Trinomial
Three term polynomial raised to the 3rd power
Pascal's Triangle
To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is just the two numbers above it added together (except for the edges, which are all "1").
x/ x+1 divided 2/x = x/x+1 * x/2 = x^2 / 2(x+1)
To divide, flip the second rational expression, multiply it with the first rational expression, and then simplify:
Quintic Binomial
Two term polynomial raised to the 5th power
addition, subtraction, and multiplication only
Under which operations are polynomials closed?
1. 10*9*8*7*6*5*4 / 7! 2. 10*9*8*7*6*5*4 / 7 * 6! 3. 10*9*8*7*6*5*4 / 7*6*5! 4. 10*9*8*6*5*4 / 6*5! 5. 10*9*8*6*5*4 / 6*5*4! 6. 10*9*8*5*4 / 5*4*3! 7. 10*9*8*4 / 4*3! 8. 10*9*8 / 3! 9. 720/3! 10. 720/ 6 11. 120
Using n!= n * (n-1)!, expand the expression and reduce until there are no more digits you can factor out. When you get down to the final fraction calculate.
i (imaginary number)
We define i as the number whose square equals -1.
The area of a tiled kitchen floor is represented by the expression 4x2 + 24x + 27, where x represents the length of a single tile. The length, l, of the floor is 9 feet more than twice the length of a tile, x. Which expression represents the width of the kitchen in terms of x?
We know that area = length • width. The area is also represented by 4x^2 + 24x + 27. We can factor this polynomial as (2x + 9)(2x + 3): 4x^2 + 24x + 27 4x^2 + 6x + 18x + 27 2x(2x + 3) + 9(2x + 3) (2x + 9)(2x + 3) Since the length of the kitchen is 9 feet more than twice an unknown, x, the length can be represented by 2x + 9. This means the width must be 2x + 3.
rth term of (x+y)^n = n(n-1)(n-2) . . . (n-r+2) x ^(n-r+1)y^(r-1) Here, n =10 and r=8. 8th term of (x+y)¹⁰ = 10(10-1)(10-2)(10-3)(10-4)(10-5)(10-6) (10-8+1) (8-1) ----------------------------------------- x y (8-1)! = 10*9*8*7*6*5*4 x³ y⁷ --------------------- 7! = 120x³y⁷
What is the 8th term of the binomial expansion (x + y)¹⁰?
The Binomial Theorem
When set equal to their expanded form, these two binomials, (x + y)2 and (x + y)3, represent the perfect square and perfect cube identities.
Closed set
When we add or subtract two polynomials, we're combining like terms, or terms with the same power of the same variable. Therefore, the exponents and variables don't change. Only the coefficients of each term might change. This means that the result must be a polynomial, and therefore polynomials are closed under subtraction.
-8x2 + 54x + 140 = -8x2 - 16x + 70x + 140 = -8x(x + 2) + 70(x + 2) = (x+ 2)(-8x+ 70)
Which expression represents -8x2 + 54x + 140 in factored form?
First consider a part of the numerator 5 + 3y2. This can be verbally described as "the sum of 5 and 3 times the square of y". Then, the entire expression in the numerator, (5 + 3y2)3, can be written as "the cube of the sum of 5 and 3 times the square of y". Next, consider a part of the denominator (4y), which can be verbally described as "the product of 4 and y". Then, the entire expression in the denominator, (4y)3, can be written as "the cube of the product of 4 and y". Dividing the two expressions yields the complete expression shown below. (5+3y^2)^3 / (4y)^3 So, the given expression can be interpreted as "the cube of the sum of 5 and 3 times the square of y divided by the cube of the product of 4 and y".
Which of the following interpretations for the given expression is correct? (5+3y^2)^3 / (4y)^3
First term of a geometric sequence
a
geometric sequence
a list of numbers for which each term can be found by multiplying the previous term by the same number.
Polynomial Long Division
a method used to divide polynomials similar to the way you divide numbers
imaginary numbers
a number that can be written as ix, where x is a real number and i is the equivalent of the square root of -1
Coefficient
a number that is multiplied by a variable in a term
trinomial
a polynomial expression consisting of three terms. Of the given choices, the polynomial x4 - 7x + 9 consists of three terms. Therefore, x4 - 7x + 9 is a trinomial.
infinite series
a series that continues on forever
finite series
a series that ends in a specific term
Closure Property
a set is closed under an operation if and only if the operation on two elements of the set produces another element of the same set
closure property
a set is closed under an operation if and only if the operation on two elements of the set produces another element of the same set; applicable to rational expressions and to rational numbers under addition, subtraction, multiplication, and division
term
a single number, a single variable, or a product of numbers and variables; in an expression, terms are related by addition or subtraction
sigma notation
a way to write an arithmetic series using the summation sign and the summand
set
a well-defined collection of distinct objects or numbers
Explicit Formula for a Geometric Sequence
a(r)^n-1
Polynomial Long Division form
a(x) / b(x) = q(x) + r(x)/b(x)
Sum of Cubes Identity
a3 + b3 = (a + b)(a2 − ab + b2)
difference of squares
a^2 -b^2 = (a+b)(a-b)
Polynomial
an expression or equation in which the terms are related to each other through addition or subtraction
polynomial
an expression or equation in which the terms are related to each other through addition or subtraction
rational expression
an expression that is the ratio of two polynomials
real number
any number that is not imaginary; the set of real numbers consists of all rational and irrational numbers
perfect square identity
a² + 2ab + b² = (a + b)²
difference of squares
a² - b² = (a + b)(a - b)
Pythagorean Theorem
a²+b²=c²
sum of cubes
a³ + b³ = (a + b)(a² - ab +b²)
extended form of the perfect square trinomial identity
a⁴ + 2a²b² + b⁴ = (a² + b²)². a² = x² and b² = 3² x⁴ + 18x² + 81 = (x²)² + (2)(x²)(3²) + (3²)² = (x²)² + 2(3²)x² + (3²)² = (x² + 3²)² = (x² + 9)²
Extended Form of Identity- Difference of Squares Identity
a⁴ − b⁴ = (a²)² − (b²)² = (a² + b²)(a² − b²)
Extended Form of Identity- sum of cubes
a⁶ + b⁶ = (a²)³ + (b²)³ = (a² + b²)[(a²)² − (a²)(b²) + (b²)²] 64x⁶ + 729 (4x²)³ + (3²)³ a² = 4x² b² = 3² = 9 64x⁶ + 729 = (4x² + 9)[(4x²)² − (4x²)(9) + 9²] 64x⁶ + 729 = (4x² + 9)(16x⁴ − 36x² + 81)
constant terms
can be written as a multiple of x to the power of 0. For example, 5 can be written as 5x0. Therefore, constants have a degree of 0.
linear terms
can be written as a multiple of x to the power of 1. For example, 6x can be written as 6x1. Therefore, linear expressions have a degree of 1.
degree 0 polynomial
constant
arithmetic division problem form
dividend/divisor = quotient + remainder/divisor
constant
fixed value; a number on its own or a letter that stands for a fixed number
common ratio
in a geometric sequence, the ratio multiplied by each term to get the next term
complex number
is the combination of a real number and an imaginary number. In standard form, we write it as a + bi, where a is the real component and bi is the imaginary component.
degree 1 polynomial
linear
imaginary numbers
n imaginary number is a number that gives a negative result when squared
common ration
r
rational expression
rational expression -> polynomial expression / polynomial expression
q(x)
represents the polynomial function in the denominator
p(x)
represents the polynomial function in the numerator.
r(x)
represents the rational function.
geometric sequence vs. geometric series
sequence = specific term series= sum of all terms in the sequence
complex numbers
the combination of a real number and an imaginary number in the form a + bi
(7x+8)³
the cube of the sum of 7x and 8
6(x − 4)^3
the difference of x and 4, cubed, times 6
degree
the largest exponent of a variable in a term of an expression, or, in expressions with terms with multiple variables, the largest sum of the exponents of a term of an expression
degree of a polynomial
the largest exponent of a variable in a term of an expression, or, in expressions with terms with multiple variables, the largest sum of the exponents of a term of an expression
6x^3 − 4
the product of x cubed and 6, minus 4
Distrubutive Property
the property stating that multiplication can be distributed over terms being added;a(b + c) = ab + ac
Associative Property of Addition
the property stating that numbers can be regrouped without changing the outcome of addition or multiplication; a + (b + c) = (a + b) + c and a(bc) = (ab)c
Communitave property
the property stating that the order in which two numbers are written does not affect the outcome of addition or multiplication;a + b = b + a and ab = ba
real numbers
the set of all rational and irrational numbers
geometric series
the sum of the terms of a geometric sequence
linear terms
two or more terms in which the variables and corresponding exponents are the same
the product of x and the sum of 6 and 8 times the square of x
x(6+8x^2)
Incorrect ex. rational expression in simplest form
x+1 / x+3 = 1/3 x(x-4) / x(x+5) = -4/5 5/ 5x + 20 = 1/ x+20 2x² / 4x = 2/4
Trinomial factoring examples
x4 − 6x2 − 27 = (x2 − 9)(x2 + 3) = (x − 3)(x + 3)(x2 + 3) x6 + 8x3 + 12 = (x3 + 6)(x3 + 2)
x²+5x+6 / x²-x-6 = (x+3)(x+2) / (x-3)(x+2) = (x+3)(x+2) / (x-3)(x+2) = (x+3) / (x-3)
x²+5x+6 / x²-x-6 1. Factor the trinomials in the numerator and the denominator. 2. Cancel out the common factors. 3. This is the simplest form of the rational expression.
simplifying the expression x²-29 / x²+x-6 - 4x-16/ x²-x-12 = x²-29 / (x+3)(x-2) - 4(x-4) / (x-4)(x+3) = x²-29 / (x+3)(x-2) - 4/x+3 = x² -29/(x+3)(x-2) - 4 / x+3 x (x-2) / (x-2) = x²-29/ (x+3)(x-2) - 4x-8 / (x+3)(x-2) = x²-4x-21 / (x+3)(x-2) = (x+3)(x-7) / (x+3)(x-2) = x-7/ x-2
x²-29 / x²+x-6 - 4x-16/ x²-x-12 1. Pull out factor 4, and factor the denominators. 2. Remove common factors 3. Find the LCM, and change to common denominator/ 4. Subtract Numerators 5. Factor the numerator 6. Remove the common factor.
Factor examples
x³ − 4x² − 9x + 36 = x²(x − 4) − 9(x − 4) = (x² − 9)(x − 4) = (x − 3)(x + 3)(x − 4) y⁵ − 81y = y(y⁴ − 81) = y[(y²)² − (3³)²] = y(y² − 9)(y² + 9) = y(y − 3)(y + 3)(y² + 9)
point slope form
y-y1=m(x-x1)
slope-intercept form
y=mx+b