Question

The polynomials $a{x}^3+3x^2-13$ and $2x^3-5x+a$ are divided by $x+2$. If the remainder in each case is the same, find the value of $a$.

Answer 1

Step1: Use the remainder theorem to evaluate the remainder when the polynomial $a{x}^3+3x^2-13$ is divided by $x+2$:

• Remainder theorem: The remainder obtained when $g\left(x\right)$ is divided by $(x-a)$ is $g\left(a\right)$, where $g\left(x\right)$ is a polynomial of degree greater than $1$ and $a$ is any real number.
• Let $f\left(x\right)=a{x}^3+3x^2-13$.
• According to the remainder theorem, the remainder obtained when $f\left(x\right)$is divided by $(x+2)$ is $f(-2)$.
• Substitute $(-2)$ for $x$ in the function $f\left(x\right)$ to obtain $f(-2)$.

$\begin{array}{rcl}f\left(x\right)& =& a{x}^3+3x^2-13\\ & \Rightarrow & f(-2)=a{(-2)}^3+3{(-2)}^2-13\\ & \Rightarrow & f(-2)=a(-8)+3\left(4\right)-13\\ & \Rightarrow & f(-2)=-8a+12-13\\ \therefore f(-2)& =& -8a-1\end{array}$

Step2: Use the remainder theorem to evaluate the remainder when the polynomial $2x^3-5x+a$ is divided by $x+2$.

• Let $g\left(x\right)=2x^3-5x+a$.
• According to the remainder theorem, the remainder obtained when $g\left(x\right)$is divided by $(x+2)$ is $g(-2)$.
• Substitute $(-2)$ for $x$ in the function $g\left(x\right)$to obtain $g(-2)$.

$\begin{array}{rcl}g\left(x\right)& =& 2x^3-5x+a\\ & \Rightarrow & g(-2)=2{(-2)}^3-5(-2)+a\\ & \Rightarrow & g(-2)=2(-8)+10+a\\ & \Rightarrow & g(-2)=-16+10+a\\ \therefore g(-2)& =& -6+a\end{array}$

Step3: Use the values of $f(-2)$ and $g(-2)$ to obtain the value of $a$.

It is given that the remainders obtained when the given polynomials are divided by $x+2$ are the same. This implies that the values of $f(-2)$ and $g(-2)$ are equivalent.

$\begin{array}{rcl}f(-2)& =& g(-2)\\ & \Rightarrow & -8a-1=-6+a\\ & \Rightarrow & 6-1=8a+a\\ & \Rightarrow & 5=9a\\ & \Rightarrow & \frac{5}{9}=a\end{array}$

Final Answer: Hence, the required value of $a$ is $\frac{5}{9}$.