Question

Use the factor theorem to determine whether v + 5 is a factor of v 4 + 16v 3 + 8v 2 - 725.

Answer 1

Not sure what "factor theorem" refers to, but one theorem it might be another name for could be the polynomial remainder theorem. It says that a polynomial
`p(x)`, when divided by a linear binomial
`x-c`, leaves a remainder whose value is
`p(c)`. If the remainder is 0, then
`x-c` is a factor of
`p(x)`.


In this case, for
`v+5=v-(-5)` to be a factor of
`p(v)=v^4+16v^3+8v^2-725`, we need to check


`p(-5)=(-5)^4+16(-5)^3+8(-5)^2-725=-1900\\eq0`

So
`v+5` is not a factor of
`v^4+16v^3+8v^2-725`.

Answer 2

By factor theorem (v+5) is not a factor of the given polynomial f(v).

Explanation:

The factor theorem states that

• f(x) has a factor (x-k) if and only if f(k) = 0

We are given the following information:

`f(v) = v^4 + 16v^3 + 8v^2 - 725`

We have to check whether (v+5) is a factor of given polynomial.

`(v+5) = (v-(-5))\\f(-5) = (-5)^4 + 16(-5)^3 + 8(-5)^2 - 725 = -1900\\f(-5) \\eq 0`

Hence, by factor theorem (v+5) is not a factor of the given polynomial f(v).