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Find the zeros of the function and state the multiplicities.

[tex]\[ g(x) = -3x^4(x + 2)^3(x + 4)^2 \][/tex]

If there is more than one answer, separate them with commas. Select "None" if applicable.

Part 1 of 2

The zero(s) of [tex]\( g \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
\square & \text{Multiplicity: } \square \\
\hline
\square & \text{Multiplicity: } \square \\
\hline
\square & \text{Multiplicity: } \square \\
\hline
\square & \text{Multiplicity: } \square \\
\hline
\end{array}
\][/tex]

None [tex]\(\square\)[/tex]

Sagot :

GinnyAnsweridn
To find the zeros of the function [tex]\( g(x) = -3x^4(x+2)^3(x+4)^2 \)[/tex], we need to set [tex]\( g(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex].

### Step-by-step Solution:

1. Identify the factors of the function:
The function [tex]\( g(x) \)[/tex] is given as:
[tex]\[ g(x) = -3x^4(x+2)^3(x+4)^2 \][/tex]
Each factor must be set to zero and solved for [tex]\( x \)[/tex].

2. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^4 = 0 \][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = 0 \][/tex]
[tex]\[ (x+2)^3 = 0 \][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = -2 \][/tex]
[tex]\[ (x+4)^2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = -4 \][/tex]

3. List the zeros:
The zeros of the function [tex]\( g(x) \)[/tex] are:
[tex]\[ 0, -2, -4 \][/tex]

### Answer for Part 1:
The zero(s) of [tex]\( g \)[/tex]:
[tex]\[ 0, -2, -4 \][/tex]
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