Question

identify the zeros, their multiplicity, and their effect on the graph of the function. 5. ( x^4 - 16x^3 + 64x^2 ) (hint: factor first)

Explanation:

Step1: Factor out common term

$x^4 -16x^3 +64x^2 = x^2(x^2-16x+64)$

Step2: Factor quadratic expression

$x^2-16x+64 = (x-8)^2$, so the full factorization is $x^2(x-8)^2$

Step3: Find zeros from factors

Set each factor equal to 0:

1. $x^2=0 \implies x=0$
2. $(x-8)^2=0 \implies x=8$

Step4: Identify multiplicity

Multiplicity matches exponent of each factor:

1. $x=0$ has multiplicity 2
2. $x=8$ has multiplicity 2

Step5: Graph effect for even multiplicity

Even multiplicity means graph touches x-axis at the zero and turns around, rather than crossing it.

Answer:

Zero	Multiplicity	Effect on Graph
$8$	$2$	Touches axis and turns around