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].
