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

$p(-5)=(-5)^4+16(-5)^3+8(-5)^2-725=-1900\neq0$

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

Answer 2

Answer:

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

Step-by-step 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.

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

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