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