5-1 Polynomial Functions

Quick Review

The standard form of a polynomial function is P ( x ) = a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 , p open x close equals , eh sub n , x to the n , plus . eh sub n minus 1 end sub . x super n minus 1 end super . plus dot dot dot plus , eh sub 1 , x plus , eh sub 0 , comma  where n is a nonnegative integer and the coefficients are real numbers. A polynomial function is classified by degree. Its degree is the highest degree among its monomial term(s). The degree determines the possible number of turning points in the graph and the end behavior of the graph.

Example

Write the polynomial function in standard form and classify it by degree. How many terms does it have? What are the possible numbers of turning points of the graph of P(x) given the degree of the polynomial?

P ( x ) = − 4 x 2 + x 4 p open x close equals negative 4 , x squared , plus , x to the fourth

Standard form arranges the terms by decreasing exponents, or P ( x ) = x 4 − 4 x 2 . p open x close equals , x to the fourth , minus 4 , x squared , .  Its degree is 4, so x 4 − 4 x 2 x to the fourth , minus , 4 x squared  is a quartic binomial. It has two terms. The graph of a quartic polynomial function can have either one or three turning points.

Exercises

Write each polynomial function in standard form, classify it by degree, and determine the end behavior of its graph.

5. y = 12 − x 4 y equals 12 minus , x to the fourth
6. y = x 2 + 7 − x y equals , x squared , plus 7 minus x
7. y = 2 x 3 − 6 x + 3 x 2 − x 4 + 12 y equals , 2 x cubed , minus 6 x plus , 3 x squared , minus , x to the fourth , plus 12
8. y = 2 x 2 + 8 − 4 x + x 3 y equals , 2 x squared , plus 8 minus 4 x plus , x cubed
9. y = 10 − 3 x 3 + 3 x 2 + x 4 y equals 10 minus , 3 x cubed , plus , 3 x squared , plus , x to the fourth
10. If the volume of a cube can be represented by a polynomial of degree 9, what is the degree of the polynomial that represents each side length?
11. A polynomial function P ( x ) p open x close  has degree n. If n is even, is the number of turning points of the graph of P ( x ) p open x close  even or odd? What can you say about the number of turning points if n is odd?

5-2 Polynomials, Linear Factors, and Zeros

Quick Review

For any real number a and polynomial P ( x ) , p open x close comma  if x − a x minus eh  is a factor of P ( x ) , p open x close comma  then a is:

• a zero of y = P ( x ) y equals p open x close
• a root (or solution) of P ( x ) = 0 , p open x close equals 0 comma  and
• an x-intercept of the graph of y = P ( x ) . y equals p open x close .

If a is a multiple zero, its multiplicity is the same as the number of times x − a x minus eh  appears as a factor.

A turning point is a relative maximum or relative minimum of a polynomial function.

Example

Find the zeros for y = 3 x 3 − 6 x 2 + 3 x y equals 3 , x cubed , minus , 6 x squared , plus 3 x  and state the multiplicity of any multiple zeros.

y = 3 x ( x 2 − 2 x + 1 ) Factor out the GCF , 3 x . y = 3 x ( x − 1 ) ( x − 1 ) Factor the quadratic . table with 2 rows and 2 columns , row1 column 1 , y equals 3 x . open . x squared , minus 2 x plus 1 . close , column 2 cap factoroutthecap gcap ccap f . comma 3 x . , row2 column 1 , y equals 3 x . open , x minus 1 , close . open , x minus 1 , close , column 2 cap factorthequadratic . . , end table

The zeros are 0, and 1 with multiplicity 2.

Exercises

Write a polynomial function with the given zeros.

12. x = − 1 , − 1 , 6 x equals negative 1 comma negative 1 comma 6
13. x = − 1 , 0 , 2 x equals negative 1 comma 0 comma 2
14. x = 1 , 2 , 3 x equals 1 comma 2 comma 3
15. x = − 2 , 1 , 4 x equals negative 2 comma 1 comma 4

Find the zeros of each function. State the multiplicity of any multiple zeros.

16. y = 3 x ( x + 2 ) 3 y equals . 3 x open x plus 2 close cubed
17. y = x 4 − 8 x 2 + 16 y equals , x to the fourth , minus , 8 x squared , plus 16
18. y = 4 x 3 − 2 x 2 − 2 x y equals , 4 x cubed , minus , 2 x squared , minus 2 x
19. y = ( x − 5 ) ( x + 2 ) 2 y equals open x minus 5 close open x plus 2 , close squared

Use a graphing calculator to find the relative maximum, relative minimum, and zeros of each function.

20. y = x 4 − 5 x 3 + 5 x 2 − 3 y equals , x to the fourth , minus , 5 x cubed , plus , 5 x squared , minus 3
21. y = 5 x 3 + x 2 − 9 x + 4 y equals , 5 x cubed , plus , x squared , minus 9 x plus 4
22. y = x 4 − 4 x − 1 y equals , x to the fourth , minus 4 x minus 1
23. y = x 3 − 3 x 2 − 3 x − 4 y equals , x cubed , minus , 3 x squared , minus 3 x minus 4

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