Question

McGraw-Hill Educa...A ALEKS - Michael A...O POLYNOMIAL AND RATIONAL FUNCTIONSThe Factor Theorem

Answer 1

The factor theorem states that, given a polynomial f(x),

`\begin{gathered} (x-a)\rightarrow\text{ is a factor of f\lparen x\rparen} \\ \Leftrightarrow \\ f(a)=0 \end{gathered}`

Thus, in our case,

`\begin{gathered} (x+2)=(x-a) \\ \Rightarrow2=-a \\ \Rightarrow a=-2 \end{gathered}`

Then, we need to evaluate P(-2) to determine whether (x+2) is a factor of P(x),

`\begin{gathered} P(-2)=(-2)^3+4(-2)^2-3(-2)-14=-8+16+6-14=0 \\ \Rightarrow P(-2)=0 \end{gathered}`

P(-2)=0

Therefore, (x+2) is a factor of P(x)