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Worked solution

Test the proposed linear factor

Determine whether x+1x+1 is a factor of the polynomial x3+x2−2x−2x^3+x^2-2x-2.

CJC · O Level revision · Example 5

Answer

Yes. x+1x+1 is a factor.

Compare your reasoning

Evaluate at the zero of the proposed factor

f(−1)=(−1)3+(−1)2−2(−1)−2=−1+1+2−2=0\begin{aligned}&f(-1)\\ & =(-1)^3+(-1)^2-2(-1)\\ &{}-2\\ & =-1+1+2-2\\ & =0\end{aligned}

Let f(x)=x³+x²−2x−2. The factor x+1 is x−(−1), so substitute −1.

Apply the factor theorem

f(−1)=0 ⟹ (x+1) is a factor of f(x)\begin{aligned}&f(-1)\\ & =0\ \Longrightarrow\ (x+1)\text{ is a factor of }f(x)\end{aligned}

A zero remainder is equivalent to divisibility by the linear factor.

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