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 .

Let’s break it down! The given polynomial is factored as \( f(x)=(x+10)^{5}(x-4)^{2}(x-8) \). From this, we can identify the zeroes. The zero occurs when each factor equals zero. First, for \( (x + 10)^{5} = 0 \), we have \( x = -10 \) with a multiplicity of 5. For \( (x - 4)^{2} = 0 \), we have \( x = 4 \) with a multiplicity of 2. Finally, for \( (x - 8) = 0 \), we get \( x = 8 \) with a multiplicity of 1. So the correct answer is \( x=-10 \), multiplicity \( 5 ; x=4 \), multiplicity \( 2 ; x=8 \), multiplicity 1. Zeroes of polynomials are fundamental in various fields including physics, engineering, and economics. When designing structures, for instance, engineers utilize polynomial zeroes to determine load distributions and stability. Understanding these zeroes helps predict how systems will react under stress and prevent disastrous failures! It’s common to mix up signs or forget about the multiplicities when determining zeroes. A tip is to carefully set each factor to zero and jot down the values alongside their corresponding powers. This way, you won’t miss any zeroes, and your zeroes list will be as neat as a well-organized desk!