Polynomial Functions Quiz Review Name: _____________________

Complete the polynomial operations. 1. (x3 + 1) + (3x2 + 5) + (9x2 – 2x) 2. (x2 – 5x + 1) – (3x2 + x – 5)

3. (2x + 7)2 4. 2 (x – 4)3

Factor the binomials. 5. x3 + 27 6. 8x3 – 1

7. 2x3 + 16 8. 64x3 – 125y3

9. 9x2 – 16 10. x2 + 100

Given the functions…find the following: f(x) = 3x – 8 g(x) = 2x2

11. g(f(5))

12. f(g(5)) 13. g(x) – f(x)

Given the functions…find the following: f(x) =

1x g(x) = x +5

14. f(g(x))

15. g f 16. g(f(x))

17. Complete the function operations.

f(x) = 2x +1 g(x) = 3x2 + x + 2

a. f(x) + g(x)

b. 16. f(x) – g(x)

f(x) = x2 g(x) = x + 3

c. f(g(5))

d. ( f g )(x )

18. Given the functions: f (x )= x 2/3 and g(x )= x 3/5

a.

f (x )i g(x ) b.

f (x )g(x )

Graph the cubic functions.

19) f (x) = x +1( )3

− 5 Initial Point: ( , )

20) f (x) = 2 x + 3( )3

− 4

Initial Point: ( , )

21) f (x) = −3 x + 4( )3

Initial Point: ( , )

22) f (x) = 1

2x3 + 4

Initial Point: ( , )

Use a graphing calculator to find the solutions to each system of equations.

23. y = x+5 y = −3 x+7 +3 24.

x+ y =5 y = − x+5( )2

25. 3x− 4y = 24 y = 2 x+ 4( )2 −9 26.

y = −(x+ 4)2 −1 y = 3 x+ 4 −5

27. Put the polynomial function into standard form: f (x )= 2x 3 − x 5 +10− x 28. Determine if the function is polynomial or not polynomial. If it is a polynomial function, circle whether it is in standard form or factored form. a. f (x )= 2x(x +1)(x −1)3

Polynomial: Yes or No Form: Standard or Factored

b. f (x )= 2x +3x − 5 Polynomial: Yes or No Form: Standard or Factored

c. f (x )= x 2 + 2x −1 +10 Polynomial: Yes or No Form: Standard or Factored

d. f (x ) = 1

3x 4 − 2x 3 − 8

Polynomial: Yes or No Form: Standard or Factored

29. Fill out the chart about end behavior.

Function Degree LC Rough Sketch (with maximum turns)

End Behavior

A f (x )=−4x3+2x +1

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

B f (x )=−10x3

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

C f (x )= 2(x +5)(x −2)

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

D f (x )= 2x7+ x 4 − x +1

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

E f (x )= x(x −3)(x +5)

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

F f (x )= 5x(x +3)2(x +1)3

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

G f (x )= 5(x +2)(x +3)3

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

H f (x )=1x 4 +2x3+1x −6

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

I f (x )= 5x 2

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

J f (x )= x(x +2)6

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

Graph Sketching: Sketch the following functions: 30. y = 2 (x + 4) (x – 2)2 Degree: ________ Leading Coefficient: ________ End behavior: x →∞, f(x)→ _____ x →−∞, f(x)→ _____ Zeros and multiplicity: _______ _______ _______

Graph Sketch

31. y = x (x + 4)3 (x – 2)2 Degree: ________ Leading Coefficient: ________ End behavior: x →∞, f(x)→ _____ x →−∞, f(x)→ _____ Zeros and multiplicity: _______ _______ _______

Graph Sketch

32. y = –3 (x – 1)2 (x + 4)3 Degree: ________ Leading Coefficient: ________ End behavior: x →∞, f(x)→ _____ x →−∞, f(x)→ _____ Zeros and multiplicity: _______ _______ _______

Graph Sketch

33. Given the factors, find the zeros. a. Factor: (x + 3) Zero: ________ d. Factor: (x – 5) Zero: ________ b. Factor: (x) Zero: ________ e. Factor: (x – 100) Zero: ________ c. Factor: (2x + 6) Zero: ________ f. Factor: (5x – 1) Zero: ________ 34. Given the zeros, find the factors. a. Zero: 9 Factor: ___________ d. Zero: –3 Factor: _____________ b. Zero: 0 Factor: ___________ e. Zero: –3i Factor: _____________

c. Zero:

35

Factor: ___________ f. Zero: 7 Factor: _____________

35. Given a zero, find another zero. a. Zero: 5i, ______________ c. Zero: 4 – 7i, ______________ b. Zero: − 17 , ______________ d. Zero: 5− 2 3 , ______________ Use your calculator to find the following information: 36. Function: y = –4x3 – 2x2 + x – 4 Degree: ____________ Lead Coeff: _____________ Rough Sketch: # of Zeros: __________ # of Real Zeros:__________ # of Imaginary Zeros: __________

37. Function: y = 3x4 + 2x2 – 3x – 7 Degree: ____________ Lead Coeff: _____________ Rough Sketch: # of Zeros: __________ # of Real Zeros:__________ # of Imaginary Zeros: __________

38. Find the polynomial with a leading coefficient of 2 that has the given zeros: 1, –2i Write f(x) in factored form: ________________________________________ Change to Standard Form: Use f(x) = 5x4 – 8x2 + 5x – 2 to answer the questions below. 39. Use substitution given x = 2. Write as an ordered pair: ____________ Write in function notation: ___________ Is the ordered pair a zero? ____________ 40. Use substitution given x = 1. Write as an ordered pair: ____________ Write in function notation: ___________ Is the ordered pair a zero? ____________ 41. Use substitution given x = –2. Write as an ordered pair: ____________

Write in function notation: ___________

Is the ordered pair a zero? ____________

!

1. Polynomial Operations Add/Subtract

1. Combine like terms. 2. Distribute any negatives, if necessary.

(x2 + 5) – (5x2 + 3x – 1)

Multiply

1. Expand. 2. Use binomial distribution. 3. Simplify.

(2x + 3)3

Factor Sum of Perfect Cubes

A3 + B3 = (A + B) (A2 – AB + B2)

x3 + 27

Factor Difference of Perfect Cubes

A3 – B3 = (A – B) (A2 + AB + B2)

8x3 – 27

2. Polynomial Functions Polynomial Functions

Sum of terms in the form: axn Cannot have negative or fractional exponents. Cannot have x in the denominator. Standard Form: f(x) = 3x5 + 10x3 – 2x + 1 Degree: The highest exponent Leading Coefficient: the coefficient of the highest exponent Factored Form: f(x) = 3x (x – 2) (5x – 3) Degree: Add the exponents of the variables Leading Coefficient: the coefficient =)

End Behavior

Even Degree Odd Degree

Positive LC

Negative LC

End Behavior Notation

Left/Right Up/Down Picture

x →∞, f(x)→∞

x →−∞, f(x)→−∞

Factors and Zeros

Given a factor, find a zero:

1. Set each factor equal to zero. 2. Solve 3. If x is a factor, 0 is a zero. 4. If 3 is a factor, ignore it!

Example: f(x) = 5 (2x + 1) (x + 4)2 Given a zero, find the factor:

1. Substitute the zero into (x – ____ ) 2. If the zero is a fraction, you can multiply

the factor by the denominator.

Graphing Polynomial Functions

1. Find the degree, leading coefficient, and end behavior.

2. Draw a small picture/sketch. 3. Find each zero. 4. Determine if the graph will bounce or

cross each zero. 5. Sketch the graph! 6. Make sure the end behavior matches

what it should be! * The number of turns is less than the degree. For example, if the degree is 5, then the function can anywhere from 1-4 turns.

Find Other Zeros

If a + bi is a zero, then a – bi is also a zero. If a + b is a zero, then a − b is also a zero.

Modeling Polynomial Functions

1. Write each zero as a factor. 2. Write the function in factored form with

the correct leading coefficient. 3. Convert to standard form by multiplying

the factors.