Exercise 18 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=4x^3-2x^2-2x

FactorOut

Factor out 2x

y=2x(2x^2-x-1)

WriteSum

Write as a sum

y=2x(2x^2-2x+x-1)

FactorOut

Factor out 2x

y=2x(2x(x-1)+(x-1) )

FactorOut

Factor out (x-1)

y=2x(x-1)(2x+1)

FactorOut

Factor out 2

y=2x(2)(x-1)(x+ 12)

Multiply

Multiply

y=4x(x-1)(x+ 12)

a+b=a-(- b)

y=4x(x-1)( x-(- 12))

WriteDiff

Write as a difference

y=4(x-0)(x-1)( x-(- 12))

Let's now use a table to organize the information. Note that, since 4≠ 0, then 4 cannot be a zero of the polynomial.

Factor	Appearances	Zero	Multiplicity
x- 0	1	0	1
x- 1	1	1	1
x-( - 1/2 )	1	- 1/2	1

We can say that 0, 1, and - 12 are zeros of multiplicity 1.