Question

Determine the multiplicity of the zeros of

the function. List the zeros, their
multiplicity (including whether the
multiplicity is odd or even) and the graph
behavior.
f(x) = (x + 3)4(x - 1)3

Answer 1

Answers:

• There are 2 unique roots
• The root x = -3 has multiplicity 4
• The root x = 1 has multiplicity 3
• The graph is shown below

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Step-by-step explanation:

The term "zeros" is the same as the term "roots".

The function f(x) = (x+3)^4(x-1)^3 has the roots x = -3 and x = 1

This is because plugging either x value into f(x) leads to f(x) = 0

We can find these roots by setting f(x) equal to 0 and solving for x

So,

f(x) = 0

(x+3)^4(x-1)^3 = 0

(x+3)^4 = 0 or (x-1)^3 = 0

x+3 = 0 or x-1 = 0

x = -3 or x = 1

The root x = -3 occurs with multiplicity 4, since this is the exponent for the factor (x+3). The root x = 1 occurs with multiplicity 3.

Even multiplicity real number roots are visually shown as "parabolic" pieces of a curve that touch at exactly one point on the x axis. Think of a ball bouncing off the floor (it goes down and then bounces back up).

Odd multiplicity real number roots are visually shown as the curve passing through the x axis.