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Find the zeros of the function and state their multiplicities.

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

If there is more than one answer, separate them with commas.

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

Part 2 of 2:
0 is a zero of multiplicity [tex]\( \square \)[/tex]
-2 is a zero of multiplicity [tex]\( \square \)[/tex]
-4 is a zero of multiplicity [tex]\( \square \)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze the function [tex]\( g(x) = -3x^4(x+2)^3(x+4)^2 \)[/tex] to find its zeros and their multiplicities.

Part 1:
From the function [tex]\( g(x) = -3x^4(x+2)^3(x+4)^2 \)[/tex], the zeros are obtained when each factor equals zero.

1. [tex]\( x^4 \)[/tex] gives a zero at [tex]\( x = 0 \)[/tex].
2. [tex]\( (x+2)^3 \)[/tex] gives a zero at [tex]\( x = -2 \)[/tex].
3. [tex]\( (x+4)^2 \)[/tex] gives a zero at [tex]\( x = -4 \)[/tex].

Thus, the zeros of the function [tex]\( g \)[/tex] are [tex]\( 0, -2, -4 \)[/tex].

Part 2:
Next, we determine the multiplicity of each zero, which is the exponent of the corresponding factor:

1. For [tex]\( x = 0 \)[/tex]:
- The factor [tex]\( x^4 \)[/tex] indicates that [tex]\( 0 \)[/tex] is a zero with multiplicity 4.

2. For [tex]\( x = -2 \)[/tex]:
- The factor [tex]\( (x+2)^3 \)[/tex] indicates that [tex]\( -2 \)[/tex] is a zero with multiplicity 3.

3. For [tex]\( x = -4 \)[/tex]:
- The factor [tex]\( (x+4)^2 \)[/tex] indicates that [tex]\( -4 \)[/tex] is a zero with multiplicity 2.

So, the final answers are:

- [tex]\( 0 \)[/tex] is a zero of multiplicity [tex]\( 4 \)[/tex].
- [tex]\( -2 \)[/tex] is a zero of multiplicity [tex]\( 3 \)[/tex].
- [tex]\( -4 \)[/tex] is a zero of multiplicity [tex]\( 2 \)[/tex].
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