MCQ
When $p(x) = x^3 + ax^2 + 2x$ +a is divided by $(x + a),$ the remainder is:
- A$0$
- B$a$
- ✓$-a$
- D$2a$
✓
Answer
Correct option: C.
$-a$
$-a$
$p(x)=x^3+a x^2+2 x+a$
$x+a=0 \Rightarrow x=-a$
By the remainder theorem, we know that when $p(x)$ is divided by $(x+a)$, the remainder is $p(-a)$.
Now,$p(-a)=x^3+a x^2+2 x+a$
$=(-a)^3+a(-a)^2+2(-a)+a$
$=-a^3+a^3-2 a+a$
$=-a$
$p(x)=x^3+a x^2+2 x+a$
$x+a=0 \Rightarrow x=-a$
By the remainder theorem, we know that when $p(x)$ is divided by $(x+a)$, the remainder is $p(-a)$.
Now,$p(-a)=x^3+a x^2+2 x+a$
$=(-a)^3+a(-a)^2+2(-a)+a$
$=-a^3+a^3-2 a+a$
$=-a$
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