Questions: For the factored polynomial function given, find all of the zeroes and their multiplicities. f(x)=(x+10)^5(x-4)^2(x-8) Select the correct answer below: x=10, multiplicity 5; x=-4, multiplicity 2; x=-8, multiplicity 1. x=-10, multiplicity 5; x=4, multiplicity 2; x=8, multiplicity 1. x=5, multiplicity 10; x=2, multiplicity 4; x=1, multiplicity 8. x=-5, multiplicity 10; x=-2, multiplicity 4; x=-1, multiplicity 8.

Solution Steps

To find the zeroes of the polynomial function and their multiplicities, identify the values of $x$ that make each factor zero. The exponent of each factor indicates the multiplicity of the corresponding zero.

The given polynomial function is $f(x) = (x + 10)^{5}(x - 4)^{2}(x - 8)$. The factors of this polynomial are $(x + 10)$, $(x - 4)$, and $(x - 8)$.

To find the zeroes, set each factor equal to zero:

1. $x + 10 = 0$ gives $x = -10$
2. $x - 4 = 0$ gives $x = 4$
3. $x - 8 = 0$ gives $x = 8$

The multiplicity of each zero corresponds to the exponent of its factor:

• For $x = -10$, the multiplicity is $5$.
• For $x = 4$, the multiplicity is $2$.
• For $x = 8$, the multiplicity is $1$.

Final Answer