Question

Use the Factor Theorem to determine whether $\text{g(x)}$ is a factor of $\text{p(x)}$ in the following case: ${\text{p(x) = x}}^3+3x^2+3x+1$, $\text{g(x) = x + 2}$

Answer 1

A polynomial is an algebraic expression in which the exponent on any variable is a whole number. Polynomial’s degree is the highest or the greatest power of a variable in a polynomial equation.

Factor Theorem

A polynomial $\text{P(x)}$ divided by $\text{Q(x)}$ results in $\text{R(x)}$ with zero remainders if and only if $\text{Q(x)}$ is a factor of $\text{P(x)}$.

Given

• Dividend is${\text{p(x) = x}}^3+3x^2+3x+1$
• Divisor is $\text{g(x) = x + 2}$

Let us put divisor is equal to zero.

$\begin{array}{rcl}g\left(x\right)& =& x+2=0\\ x& =& -2\end{array}$

∴ Zero of $\text{g(x)}$ is $-2$

Let us consider

Let${\text{p(x) = x}}^3+3x^2+3x+1$

put $\text{x = -2}$ in the above equation

$\begin{array}{rcl}p(-2)& =& (-2)3+3(-2)2+3(-2)+1\\ p(-2)& =& –8+12–6+1\\ p(-2)& =& -1\ne 0\end{array}$

Since reminder is not equal to zero, so $x+2$ is not a factor of ${\text{x}}^3+3x^2+3x+1$.

∴ By factor theorem, $\text{g(x)}$ is not a factor of $\text{p(x)}$.