Exercise 17 Page 348

We want to determine the zeros of the given polynomial function as well as their multiplicity. We say that b is a zero of multiplicity n when (x- b) appears n times as a factor of a polynomial. Let's rewrite the given function in order to find the zeros.

y=x^4-8x^2+16

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Factor the trinomial

SplitIntoFactors

Split into factors

y=x^4-2(x^2)(4)+16

SplitIntoFactors

Split into factors

y=x^(2* 2)-2(x^2)(4)+16

ProdInExponent

a^(m* n)=(a^m)^n

y=(x^2)^2-2(x^2)(4)+4^2

FacNegPerfectSquare

a^2-2ab+b^2=(a-b)^2

y=(x^2-4)^2

WritePow

Write as a power

y=( x^2-2^2 )^2

FacDiffSquares

a^2-b^2=(a+b)(a-b)

y=( (x+2)(x-2) )^2

PowProd

(a * b)^m=a^m* b^m

y=(x+2)^2(x-2)^2

a+b=a-(- b)

y=(x-(- 2))^2 (x-2)^2

Let's now use a table to organize the information.

Factor	Appearances	Zero	Multiplicity
x-( - 2)	2	- 2	2
x- 2	2	2	2

We can say that - 2 and 2 are zeros of multiplicity 2.