Question

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

• A. $(x+1)$ and $(x^2+x+1)$
• B. $(x+1)$ and $(x^2-x+1)$
• C. $(2x+1)$ and $(x^2+x+1)$
• D. $(x+1)$ and $(x^2+3x+1)$

Answer 1

The correct option is A $(x+1)$ and $(x^2+x+1)$

Let $p\left(x\right)=(x^3+2x^2+2x+1)$ be the given polynomial.

Substitute $(x=-1)$:
$p(-1)=(-1{)}^3+2\times (-1{)}^2+2\times (-1)+1$
$p(-1)=0$
$\therefore x=-1$ is a root of $p\left(x\right)$
$\Rightarrow (x+1)$ is a factor of $p\left(x\right)$ by factor theorem.

To find the other factor, divide $p\left(x\right)$ 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 $(x^2+x+1)$.