Question

Find the zeros of the polynomial function and state the multiplicity of each.

f(x) = 5(x + 8)^2(x - 8)^3

Answer 1

x+8=0,x=-8
hence x=-8 is a zero and multiplicity is 2
similarly find others.

Answer 2

Because you have this polynomial written in factored form:
\[f(x) = 5(x+8)^2(x-8)^3 = 5(x+8)(x+8)(x-8)(x-8)(x-8)\]you can use the zero product property of multiplication to find the zeros. Every value of \(x\) that makes one of those product terms = 0 is going to be a zero of the polynomial. As surjithayer demonstrated, set each one equal to 0 and solve for the value of \(x\):
\[x+8 = 0\]\[x = -8\]is a zero of the polynomial. The multiplicity is just the number of times that particular zero is found. As the factor which provides it has an exponent of 2, the multiplicity is 2. The multiplicities of all of the roots must add up to the highest exponent of the polynomial when expanded. Thank your lucky stars that you get to do this problem with the factored polynomial, instead of the expanded polynomial, which is

\[f(x) =5 x^5-40 x^4-640 x^3+5120 x^2+20480 x-163840\]

If we plug in \(x=-8\) we can see that \(f(-8) = 0\):

\[5(-8)^5-40(-8)^4-640(-8)^3+5120(-8)^2+20480(-8)-163840 =\]\[\qquad 5(-32768) - 40(4096)-640(-512)+5120(64)+20480(-8)-163840 = \]\[\qquad -163840-163840+327680+327680-163840 -163840\]\[\qquad = 0\]

And the highest exponent is 5, so we'll have a total of 5 zeros (though only 2 unique values, as they both have multiplicities > 1)