Question

Find the factors of $x^3+2x^2+2x+1$.

• A. a must be 2
• B. a can take only three values
• C. a can be any integer
• D. a is a negative integer

Answer 1

Answer: (A), (B)

The correct options are
A

a must be 2

B

a can take only three values

Let $p\left(x\right)=(x^3+2x^2+2x+1)$ be the given polynomial.
By observation, we can see that by putting $(x=-1)p\left(x\right)=0$
$\therefore x=-1$ is the root of $p\left(x\right)$
$\Rightarrow (x+1)$ is a factor of $p\left(x\right)$
To find the other factor, we perform long division of $p\left(x\right)$ by $(x+1)$,
$x+1\stackrel{x^2+x+1}{\sqrt{x^3+2x^2+2x+1}}{x}^3+x^2--\overline{}{x}^2+2x{x}^2+x--\overline{}x+1x+1--\overline{}0$
$\therefore$ The two factors of $p\left(x\right)$ are $(x+1)$ and $(x^2+x+1)$.