Prentice Hall Algebra 2

Chapter 5 Polynomials and Polynomial Functions > 5-2 Polynomials, Linear Factors, and Zeros

You can write the polynomial functions in Problem 3 in factored form as f ( x ) = ( x + 2 ) ( x − 2 ) ( x − 3 ) f open x close equals open x plus 2 close open x minus 2 close open x minus 3 close and g ( x ) = ( x + 2 ) 2 ( x − 2 ) ( x − 3 ) . g open x close equals open x plus 2 , close squared , open x minus 2 close open x minus 3 close . In g open x close the repeated linear factor x plus 2 makes negative 2 a multiple zero.

In particular, since the linear factor x plus 2 appears twice, you can say that negative 2 is a zero of multiplicity 2. In general, a is a zero of multiplicity n means that x minus eh appears n times as a factor.

Take Note Key Concept: How Multiple Zeros Affect a Graph

If a is a zero of multiplicity n in the polynomial function y equals p open x close comma then the behavior of the graph at the x-intercept a will be close to linear if n equals 1 comma close to quadratic if n equals 2 comma close to cubic if n equals 3 comma and so on.

Problem 4 Finding the Multiplicity of a Zero

What are the zeros of f ( x ) = x 4 − 2 x 3 − 8 x 2 ? f open x close equals , x to the fourth , minus , 2 x cubed , minus , 8 x squared , question mark What are their multiplicities? How does the graph behave at these zeros?

f ( x ) = x 4 − 2 x 3 − 8 x 2 = x 2 ( x 2 − 2 x − 8 ) Factor out the GCF , x 2 . = x 2 ( x + 2 ) ( x − 4 ) Factor ( x 2 − 2 x − 8 ) . table with 3 rows and 3 columns , row1 column 1 , f , open x close , column 2 equals , x to the fourth , minus 2 , x cubed , minus 8 , x squared , column 3 , row2 column 1 , , column 2 equals , x squared . open . x squared , minus 2 x minus 8 . close , column 3 cap factoroutthecap gcap ccap f . comma , x squared , . , row3 column 1 , , column 2 equals , x squared . open , x plus 2 , close . open , x minus 4 , close , column 3 cap factor . open . x squared , minus 2 x minus 8 . close . . , end table

Since x 2 = ( x − 0 ) 2 , x squared , equals open x minus 0 , close squared , comma the number 0 is a zero of multiplicity 2. The numbers negative 2 and 4 are zeros of multiplicity 1.

The graph looks close to linear at the x-intercepts negative 2 and 4. It resembles a parabola at the x-intercept 0.

A w-shaped curved is closed to linear at the x-intercept (negative 2, 0), close to a parabola at the x-intercept (0, 0), and close to linear at the x-intercept (4, 0).

✓ Got It?

What are the zeros of f ( x ) = x 3 − 4 x 2 + 4 x ? f open x close equals , x cubed , minus , 4 x squared , plus 4 x question mark

What are their multiplicities? How does the graph behave at these zeros?

If the graph of a polynomial function has several turning points, the function can have a relative maximum and a relative minimum. A relative maximum is the value of the function at an up-to-down turning point. A relative minimum is the value of the function at a down-to-up turning point.

An N-shaped graph has two turning points. The graph falls to a vertex in quadrant 3, the graph’s relative minimum, and then rises to a vertex in quadrant 1, its relative maximum, before falling through quadrant 4.

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Take Note Key Concept: How Multiple Zeros Affect a Graph

Problem 4 Finding the Multiplicity of a Zero

Think

✓ Got It?

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