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

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23-05-2018
Mathematics

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

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LammettHash LammettHash · Answer 1 · 2018-05-23T20:59:36+02:00

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 [tex]p(x)[/tex], when divided by a linear binomial [tex]x-c[/tex], leaves a remainder whose value is [tex]p(c)[/tex]. If the remainder is 0, then [tex]x-c[/tex] is a factor of [tex]p(x)[/tex].

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

[tex]p(-5)=(-5)^4+16(-5)^3+8(-5)^2-725=-1900\neq0[/tex]

So [tex]v+5[/tex] is not a factor of [tex]v^4+16v^3+8v^2-725[/tex].

ChiKesselman ChiKesselman · Answer 2 · 2019-10-30T06:27:57+01:00

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:

[tex]f(v) = v^4 + 16v^3 + 8v^2 - 725[/tex]

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

[tex](v+5) = (v-(-5))\\f(-5) = (-5)^4 + 16(-5)^3 + 8(-5)^2 - 725 = -1900\\f(-5) \neq 0[/tex]

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