QuestionFind the zeros and their multiplicities for the function: $y=4 x^{3}(x+2)^{3}(x+1)$ .

Studdy Solution

STEP 1

Assumptions1. The given function is $y=4x^{3}(x+)^{3}(x+1)$ . We are asked to find the zeros and their multiplicity

STEP 2

The zeros of a function are the values of $x$ that make the function equal to zero. In other words, we need to solve the equation $y=0$ for $x$ .
$0=4x^{}(x+2)^{}(x+1)$

STEP 3

We can see that the equation is already factored, so we can set each factor equal to zero to find the zeros.
$x^{3}=0, (x+2)^{3}=0, (x+1)=0$

STEP 4

olving each equation for $x$ gives us the zeros of the function.
$x=0, x=-2, x=-1$

STEP 5

The multiplicity of a zero is the number of times it appears as a root, which is indicated by the exponent of the factor.For $x=0$ , the factor is $x^{3}$ , so the multiplicity is3. For $x=-2$ , the factor is $(x+2)^{3}$ , so the multiplicity is3. For $x=-1$ , the factor is $(x+1)$ , so the multiplicity is1.
So the zeros of the function $y=4x^{3}(x+2)^{3}(x+1)$ are $0$ with multiplicity $3$ , $-2$ with multiplicity $3$ , and $-1$ with multiplicity $1$ .

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