Question

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

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 $2x^3+x^2–2x–1$
• Divisor is $x+1$

Let us put divisor is equal to zero.

$\begin{array}{rcl}g\left(x\right)& =& x+1=0\\ x& =& -1\end{array}$

∴ Zero of $\text{g(x)}$ is $-1$

Let us consider

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

put $\text{x = -1}$in the above equation

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

∴ Reminder $\text{p(-1) = 0}$

Since reminder is zero, so $\text{x + 1}$ is a factor of $2x^3+x^2–2x–1$.

Hence, by factor theorem, $\text{g(x)}$ is a factor of $\text{p(x)}$.