Precalculus: Polynomial and Rational Functions

What is the vertex of y=(0.5x+5)²-9

(-10,-9)

An equation/ graph is a function if..

1. It passes Vertical Line Test (VLT). Each x must have 1 y solution. 2. The exponential power of y is not even 3. y is not inside an absolute value ||

HOW DO WE ANSWER THIS: Using long division: What is 6x²+7x-20 divided by 2x+5

1. Set up the problem: 2x+5 ⟌(6x²+7x-20) 2. 2x goes into 6x² 3x times. Multiply 3x by 2x+5. 3x 2x+5 ⟌(6x²+7x-20) -6x²-15x ----------------------------- -8x 3. Keep doing it until you ended up with a remainder (0)

HOW DO WE ANSWER THIS: Find the zeros of x²+3x-10

Methods of finding zeros (use the easiest possible one): 1. Factor in the expression. 2. Completing squares (recommended only if b is even) 3. If 1 doesn't work, use the quadratic formula Let's use Method #1 0=x²+3x-10 0=(x+5)(x-2) x= -5 , 2

Is this set a function: (-5,3) , (-4,9) , (-3,9) , (-3,4)

No

What are the solutions to: {2,8,12,15} n {3,12,15,16}

{12,15}

What are the solutions to: {2,8} U {3,6,8,10}

{2,3,6,8,10}

What are the solutions to: {2,6,8,12,15} n {3,4,18,22,28}

ø

What is the vertex of y=(3x+12)²+9

(-4,9)

What is the vertex of y=(x+5)²-9

(-5,-9)

What is the vertex of y=8(x+5)²-9

(-5,-9)

Difference of Square Formula

(a²+b²)=(a+b)(a-b)

Find the zeros of 2x²+4x-9 by completing the square.

-1±√(11/2)

What are the zeros of x²+4x+9

-2 ± i√5

Find the zeros of 3x² -16x -12

-2/3 and 6

Find the average rate of change over the interval [3,4] in f(x)= 3x⁴-8x³-37x²+2x+40

-28

Using difference of cubes, find the zeros of 8x³+27

-3/2

Find the zeros of x²+3x-10

-5 and 2

Polynomial synthetic division

-Easier than polynomial long division -You can only do f(x) / (x-n). Divide any polynomial functions by n, a zero of x-n. If the remainder is a zero, then n is one of the zeros of f(x).

Polynomial functions

-function with variables raised to power higher than 1 - represented as f(x)=axⁿ+bxⁿ⁻¹+cxⁿ⁻²+...+z

Using RZT, what are the possible zeros of 2x⁴-6x³+2x-8

1, 1/2 , 2 , 4, 8, -1, -1/2, -2 ,-4, -8

Horizontal Asymptotes of rational functions

1. A rational function has a horizontal asymptote at y=0 if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. 2. A rational function has a horizontal asymptote at y=a/b if the degree of the polynomial in numerator is equal to the degree of the polynomial in denominator. The leading coefficient of polynomial in the numerator and denominator are a and b respectively. 3. A rational function has no horizontal if the degree of the polynomial in the numerator is greater than the degree of the polynomial in denominator. A rational function cannot have a horizontal asymptote if it already have an oblique asymptote.

Vertical Asymptotes of rational functions

1. A rational function, f(x)/g(x), has a vertical asymptote at x=a when g(a)=0.

Rules for imaginary zeros of polynomial

1. All polynomials must have an even number of imaginary zeros 2. If a+bi is a zero of a polynomial, then its conjugate, a-bi, must also be a zero.

Injective function

1. Also known as one-to-one function. It denotes the mapping of two sets. Two values of x cannot have the same value of y. 2. Passes the Horizontal and Vertical Line Test. f(x)=3x+5 is an example of injective function. f(x)=x² is not an injective function.

Oblique (slant) Asymptotes of rational functions

1. An oblique asymptote is written in the form of y=mx+b 2. A rational function can only have oblique asymptote if the degree of a polynomial in the numerator is one more than the degree of polynomial in the denominator. 3. To find the equation of an oblique asymptote, divide the polynomial in the numerator by the polynomial in the denominator. Ignore the remainder and do not add it to your answer.

HOW DO WE ANSWER THIS: Using synthetic division, calculate (x³-2x²-5x+6)÷(x-3)

1. Bring the value of leading coefficient down and multiply it by 3. Subtract that number from the next leading coefficient and bring down the answer. Repeat until you end up with a remainder. 3 {1 -2 -5 6} ↓ 3 3 -6 1 1 -2 0 2. Because the remainder is 0, we can also conclude that 3 is one of the zeros of x³-2x²-5x+6.

HOW DO WE ANSWER THIS: Using synthetic division, calculate (3x³-2x²-150)÷(x-4)

1. Bring the value of leading coefficient down and multiply it by 4. Subtract that number from the next leading coefficient and bring down the answer. Repeat until you end up with a remainder. 4{3 -2 0 -150} ↓ 12 40 160 3 10 40 10 2. We got 3x²+10x+40 with a remainder of 10. Remember, if we have a remainder, add (remainder/divisor) to your answer. 3. In conclusion we have 3x²+10x+160+ [10/(x-4)]. We can also conclude that 4 is NOT a zero of 3x³-2x²-150.

Domain, Range, and End Behavior of Rational Function.

1. Depended on the asymptotes. 2. EXAMPLE: The asymptotes of 1/x are x=0 and y=0 3. Domain and Range of 1/x: Domain: (-∞, 0) U (0,∞) Range: (-∞, 0) U (0,∞) 4. End Behavior of 1/x As x→-∞, f(x)→0 As x→-∞, f(x)→0 As x→0⁺, f(x)→∞ As x→0⁻, f(x)→-∞

How do we find the zeros of a quadratic function?

1. Factor in the expression. 2. Completing squares (recommended only if b is even) 3. If 1 doesn't work, use the quadratic formula

How do we graph quadratic equations?

1. Find the vertex 2. Find the y-intercept (when x=0) 3. Find other points if necessary

For f(x)= 3 / (x+3) 1. Find all asymptotes 2. Find the domain and range 3. Find the End Behavior

1. The asymptotes are x=-3 and y=0 2. Domain: (-∞,-3) U (-3, ∞) Range: (-∞,0) U (0,∞) 3. As x→∞, f(x)→0 As x→-∞, f(x)→0 As x→-3⁺, f(x)→∞ As x→-3⁻, f(x)→-∞

HOW DO WE ANSWER THIS: Using difference of cubes, factor 8x³-64

1. The formula for difference of cubes is: a³±b³ = (a±b)(a²∓ab+b²) 2. Factor out the 8: 8(x³-8) 3. Find the value of a and b: x and 2 respectfully. 4. Substitute x and 2 for a and b for (a±b)(a²∓ab+b²): = 8(x³-8) =8 (x±2)(x²∓x(2)+(2)²) =8 (x-2) (x² + 2x +4) (x² + 2x +4) cannot be factor out further so 8 (x-2) (x² + 2x +4) is the answer in the simplest term.

HOW DO WE ANSWER THIS: Using difference of cubes, find the zeros of 8x³+27

1. The formula for difference of cubes is: a³±b³ = (a±b)(a²∓ab+b²) 2. The value of a and b are 2 and 3 respectfully. 3. Solve 8x³+27 = (2x+3) (4x²-6x+9) 8x³+27 = (2x+3) (4x²-6x+9) 4x²-6x+9 has no solution because (-6)² is less than 4(4)(9). However, the solution to 2x+3=0 is -3/2, making it the only solution to 8x³+27

Rational Zero Theorem (RZT)

1. This theorem shows the possible rational zeros of a polynomial. (factor of constant / factor of leading coefficient) are the possible rational zeros. 2. For axⁿ+bxⁿ⁻¹+cxⁿ⁻²+...+z, ±(all factors of z/all factors of a) are the possible rational zeros of the polynomial. 3. This theorem will not work on irrational and imaginary numbers.

HOW DO WE ANSWER THIS: Find all zeros of f(x)=3x⁴-8x³-37x²+2x+40

1. Use Descartes Rule of Sign if necessary 2. Use RZT. According to RZT, ±(factors of 40 / factors of 3), are all the possible zeros of f(x)=3x⁴-8x³-37x²+2x+40. This includes ±5/3, ±20, and ±8/3. 3. Choose and test any RZT. Let's choose -2. f(-2) will equal in 0. If we use synthetic division, we can conclude that (3x⁴-8x³-37x²+2x+40) / (x+2) is equal to 3x³-14x²-9x+20 and has a remainder of 0. Therefore, -2 is one of the zeros. 4. Use RZT again for 3x³-14x²-9x+20 and test random number. Let's choose 1. If we do (3x³-14x²-9x+20) / (x-1), we will get 3x²-11x-20 when a remainder of 0. 5. After using the quadratic formula for 3x²-11x-20, we should get x=-4/3 and x=5. 6. SOLUTIONS: 3x⁴-8x³-37x²+2x+40 = (3x³-14x²-9x+20)(x+2) = (3x²-11x-20)(x-1)(x+2) =(x+4/3)(x-5)(x-1)(x+2)

How do we use RZT to find the zeros of polynomials

1. Use RZT to find the possible rational zeros of the polynomial, f(x) 2. If "n" is a possible rational zero, divide f(x) by (x-n) using synthetic division. 3. If f(n)=0 and f(x)/(x-n) has no remainder, then n is a zero of a polynomial. If not, check with other possible zeros 4. Repeat the steps if necessary by evaluating the smaller polynomials. For quadratic equations, use the quadratic formula.

Difference quotient

1. Used in calculus 2. Represent the average rate of change between x=a and x=a+b 3. The formula is [f(a+h)-f(a)] / h

Inverse function

1. a function that undoes the operation of f 2. If g is the inverse of f and f(x)=y, then g(y)=x. 3. The inverse of y=3x+9 is y=(9-x)/3

If f(x)=x+2 and g(x)=3x, what is (f∘g)(5)

17

Using long division: What is 2x²+7x-4 divided by x-3?

2x + 13 + [35/(x-3)]

Write 3x²+6x-7 to vertex form.

3(x+1)²=10

Using long division: What is 6x²+7x-20 divided by 2x+5

3x-4

Using synthetic division, calculate (3x³-2x²-150)÷(x-4)

3x²+10x+160+ [10/(x-4)]

SHOW WORK: Write 3x²+6x-7 to vertex form.

3x²+6x-7 = 0 3x²+6x = 7 3x²+6x+3 = 10 3(x²+2x+x)-10=0 3(x+1)²-10 = 0

f(x) is a polynomial. Find the difference quotient for f(x)=2x²+3x-1

4a+2h+3

Find all zeros of f(x)=3x⁴-8x³-37x²+2x+40

5, 1, -4/3, -2

What are the zeros of x²+36

6i and -6i

Using difference of cubes, factor 8x³-64

8(x-2) (x²+2+4)

Composite functions

A function inside another function. Represented as (f∘g) (x) or f(g(x))

Rational Functions

A ratio of 2 polynomials. f(x)/g(x) is a rational function if f(x) and g(x) are both polynomials.

Quadratic Functions

A second degree polynomial.

cubic function

A third degree polynomial Represented as ax³+bx²+cx+d

What do we do with a remainder when dividing 2 polynomials?

Add (remainder/divisor) to your answer

Quadratic Formula

Can be use to find the zeros of any quadratic equations. Just substitute the coefficient for a, b, and c.

complex number

Combination of imaginary and real number. Usually represented as a+bi when a is real and bi is imaginary.

How do we convert standard quadratic equation to vertex form?

Complete the square by using the formula [(b/2)²] / a. Then, move all the terms to one side.

HOW DO WE ANSWER THIS: Using long division: What is 2x²+7x-4 divided by x-3?

Divide two polynomials the same way you would divide 2 numbers. For the first step, we know that x goes into 2x² 2x times. After dividing, we ended up with 2x+13 with a remainder of 35. Remember, if we have a remainder, add (remainder/divisor) to your answer. Add 2x+13 to 35 / (x-3)

What are the domain, range, and end behavior of 53x⁵-9x⁴+9x³-8x²+9x-1

Domain: (-∞,∞) Range: (-∞,∞) End Behavior: f(x)→ -∞ , as x→ -∞ f(x)→ ∞ , as x→ ∞

What are the domain, range, and end behavior of -3x¹¹-10x⁸-2

Domain: (-∞,∞) Range: (-∞,∞) End Behavior: f(x)→ ∞ , as x→ -∞ f(x)→ -∞ , as x→ ∞

What are the domain, range, and end behavior of 3x⁶-10x⁴-8x²+9x-1

Domain: (-∞,∞) Range: (-∞,∞) End Behavior: f(x)→ ∞ , as x→ -∞ f(x)→ ∞ , as x→ ∞

Hole in rational function

Holes occur at x=a in f(x)/g(x) when f(a)/g(a)=0. It may pass the asymptote

End behavior of an odd degree polynomial

If the leading coefficient is positive: As x approaches -∞, y will approach to -∞. As x approaches ∞, y will approach to ∞. If the leading coefficient is negative: As x approaches -∞, y will approach to ∞. As x approaches ∞, y will approach to -∞.

End behavior of an even degree polynomial

If the leading coefficient is positive: As x approaches -∞, y will approach to ∞. As x approaches ∞, y will approach to ∞. If the leading coefficient is negative: As x approaches -∞, y will approach to -∞. As x approaches ∞, y will approach to -∞.

Sketch, compression, and reflection: y=a(bx-h)²+k

If |a|>1, it is a vertical stretch If 0<|a|<1, it is a vertical compression If a<0, it is a reflection If b>1, it is a horizontal compression If 0<b<1, it is a horizontal stretch

A n B

Intersection: A set that includes A OR B

HOW DO WE ANSWER THIS: Find the zeros of 3x² -16x -12

Methods of finding zeros (use the easiest possible one): 1. Factor in the expression. 2. Completing squares (recommended only if b is even) 3. If 1 doesn't work, use the quadratic formula Let's use method #1 3x² -16x -12 Can't factor out common number. For ax² +bx +c, when a>1, find 2 numbers that add up to b but also multiply to ac. 0=3x² -16x -12 0=3x² -18x+ 2x -12 0=3x(x-6) +2(x-6) 0=(3x+2)(x-6) x= -2/3 and 6

Is √y+9=8 a function?

No

Is|y|=x+2 a function?

No

Descartes Rule of Sign

Number of possible positive, negative, and imaginary zeros based on the number of sign (+/-) changes. Possible zeros = # of changes OR any 2 less than # of changes

How do we use Descartes Rule of Sign? Show example.

Positive Zeros: Count the number of sign changes. The number of possible positive zeros is equal or any 2 less than the number of sign changes in f(x). For example, ax⁵-bx⁴-cx³+dx²-ex+f. From ax⁵ to bx⁴ it changes from + to -. From bx⁴ to cx³, it stays the same. There are 4 sign changes so there are 2 or 4 possible positive zeros. Negative Zeros: The number of possible negative zeros is equal or any 2 less than the number of sign changes in f(-x). Count the number of sign changes of f(-x). For example, if f(x)= ax⁵-bx⁴-cx³+dx²-ex+f, then f(-x) would be -ax⁵-bx⁴+cx³+dx²+ex+f. There are 1 sign changes so there are 1 possible negative zeros. Imaginary Zeros: Because ax⁵-bx⁴-cx³+dx²-ex+f is a fifth degree polynomial, it can only have 5 or less zeros in total. Using the given information above, we can conclude that there are 0 or 2 possible imaginary zeros. All polynomials must have an even number of imaginary zeros

How many possible positive, negative, and imaginary zeros does f(x)=x³-4x²+3x+8 have? DO NOT EVALUATE.

Possible Positive Zeros: 2 or 0 Possible Negative Zeros: 1 Possible Imaginary Zero: 0 or 2

How many possible positive, negative, and imaginary zeros does f(x)=-2x ⁴+x³-6x²-7x+1 have? DO NOT EVALUATE

Possible Positive Zeros: 3 or 1 Possible Negative Zeros: 1 Possible Imaginary Zeros: 0 or 2

Range and domain of all polynomial functions

Range: All real number or (-∞,∞) Domain: All real number or (-∞,∞)

Range and domain definition

Range: all y solutions of a function Domain: all x solutions of a function ( ) is same as <> [] is same as ≤≥

Polynomial Long Division

Similar to dividing 2 numbers. Divide a polynomial by another polynomial with lower degree.

Graphing rational functions

Substitute several numbers of x for solve for y. Plot those numbers and draw the graph as close as possible to each asymptotes without touching it.

HOW DO WE ANSWER THIS: f(x) is a polynomial. Find the difference quotient for f(x)=2x²+3x-1

The formula for difference quotient is [f(a+h)-f(a)] / h. Start plugging in a+h and h and simplify the expression. {2(a+h)²+3(a+h)-1 -[2(a)²+3(a)-1]} / h ={2a²+4ah+2h²+3a+3h-1-2a²-3a+1} / h =4ah+2h²+3h / h =4a+2h+3

Discriminant Formula Rule

This formula is part of a quadratic formula. If d=b²-4ac: The quadratic equation has 2 real solutions if d>0 The quadratic equation has 1 real solution if d=0 The quadratic equation has 2 imaginary solutions if d<0

HOW DO WE ANSWER THIS: Find the zeros of 2x²+4x-9 by completing the square.

To complete the square, move the c to the other side: 2x²+4x-9 = 0 2x²+4x = 9 Add " [(b/2)²] / a " to both side and solve for x 2x²+4x + 2 = 9 +2 2(x²+2x + 1) = 11 2(x+1)² = 11 (x+1)² = 11/2 x+1 = ±√11/2 x= ±√11/2 -1

A U B

Union: A set that includes both A AND B

HOW DO WE ANSWER THIS: Write the equation of all asymptotes of f(x)= [x²-6x+8] / [-4x+4]

Vertical Asymptote: The denominator must equal to zero. When x=1, -4x+4 is equal to zero. Horizontal Asymptote: There are no horizontal asymptote because the degree of the polynomial in the numerator is greater than the degree of the polynomial in denominator. Oblique Asymptote: There is an oblique asymptote. [x²-6x+8] divided by [-4x+4] is -0.25x + 1.25 with a remainder of 3. Ignore the remainder. -0.25x + 1.25 is the oblique asymptote.

HOW DO WE ANSWER THIS: Write the equation of all asymptotes of f(x)= [x²+5x+6] / [x²+x-2]

Vertical Asymptote: The denominator must equal to zero. When x=1, x²+x-2 or (x-1)² is equal to zero. Horizontal Asymptote: Because both polynomial are second degree, the asymptote must the ratio of 2 leading coefficient. That would be 1/1. Oblique Asymptote: There are no oblique asymptote because there are horizontal asymptote.

HOW DO WE ANSWER THIS: Find a 3rd degree polynomial function with a real coefficients and has a roots of 6 and -5+2i. (2,-636) is also a solution to that function.

We are given that the zeros are 6 and -5+2i. The conjugation of -5+2i, -5-2i, must also be a zero because a polynomial cannot have an odd number of imaginary zeros. Let "a" be the factor: a(x-6)(x+5-2i)(x+5+2i) =a(x-6)(x²+5x+2ix+5x+25+10i-2ix-10ix-4i²) =a(x-6)(x²+5x+2ix+5x+25+10i-2ix-10ix+4) =a(x-6)(x²+10x+29) =a(x³+4x²-31x-174) (2,-636) is a solution. Set the equation equal to -636 and solve for a. 636=a(2³+4(2)²-31(2)-174) 636=a(-212) a = 3 Factor 3 to x³+4x²-31x-174 to find your answer.

Is this set a function: (-5,3) , (-4,9) , (-3,9) , (-2,4)

Yes

Is y+y³-9=x a function?

Yes

Quadratic Function: Completing square formula

[(b/2)²] / a

Asymptotes

a line or curve that acts as the limit of another line or curve. In rational functions, there are horizontal, vertical, and oblique asymptotes. The graph can cross a horizontal and oblique asymptote but never vertical asymptote.

Difference of cubes formula

a³±b³ = (a±b)(a²∓ab+b²)

Discriminant Formula

b²-4ac

Average rate of change between 2 points of nonlinear function

f(a)-f(b) / a-b

Find a 3rd degree polynomial function with a real coefficients and has a roots of 6 and -5+2i. (2,-636) is also a solution to that function.

f(x)=3x³+12x²-93x-522

imaginary number

represented as i when i=√(-1)

Highest degree of a polynomial

sum of the most exponential power Example: 3ab⁵+6b⁴-9ab+9 is a 6th degree polynomial.

Write the equation of all asymptotes of f(x)= [x²-6x+8] / [-4x+4]

x=1 y= -0.25x + 1.25

Using synthetic division, calculate (x³-2x²-5x+6)÷(x-3)

x²+x-2

Using long division: What is 3x³-5x²+10x-3 divided by 3x+1?

x²-2x+4 + [-7 / [3x+1)]

Write the equation of all asymptotes of f(x)= [x²+5x+6] / [x²+x-2]

y=1 x=1

Quadratic vertex form

y=a(bx-h)²+k a is the vertical stretch and compression. Reflection is less than 0. b is the horizontal stretch and compression h is the x transformation k is the y transformation (-h,k) is the vertex

What are the solutions to: {0,9,12,13} u ø

{0,9,12,13}