Question

How do you use the remainder theorem to find $P\left( a \right)$
$P\left( x \right)=6{{x}^{3}}-{{x}^{2}}+4x+3,a=3$?

Answer 1

Hint: To solve this question, we need the statement of the remainder theorem. The remainder theorem states that when a polynomial $p\left( x \right)$ is divided by $x-a$, then the remainder of the division is equal to $p\left( a \right)$. In the above question, we have been given a polynomial $P\left( x \right)$ as $P\left( x \right)=6{{x}^{3}}-{{x}^{2}}+4x+3$ and are asked to evaluate $P\left( a \right)$, where a is given to be equal to $3$. Therefore, we wil divide the given polynomial by $x-3$ and the remainder obtained from the division performed will give the required value of $P\left( a \right)$.

Complete step-by-step answer:
The polynomial given in the above question is
$\Rightarrow P\left( x \right)=6{{x}^{3}}-{{x}^{2}}+4x+3$
According to the question, we need to calculate $P\left( a \right)$, where $a=3$ using the remainder theorem. Therefore, effectively we need to calculate the value of $P\left( 3 \right)$. From the remainder theorem we know that on dividing a polynomial $p\left( x \right)$ by $x-a$, the value of the remainder will be equal to $p\left( a \right)$. Therefore, for calculating the value of $P\left( 3 \right)$, we will divide the polynomial $P\left( x \right)$ by $x-3$ as shown below.
\[x-3\overset{6{{x}^{2}}+17x+55}{\overline{\left){\begin{align}
  & 6{{x}^{3}}-{{x}^{2}}+4x+3 \\
 & \underline{6{{x}^{3}}-18{{x}^{2}}} \\
 & 17{{x}^{2}}+4x \\
 & \underline{17{{x}^{2}}-51x} \\
 & 55x+3 \\
 & \underline{55x-165} \\
 & \underline{168} \\
\end{align}}\right.}}\]
From the above division, we got the remainder equal to $168$. Therefore, by the remainder theorem we can say that the value of $P\left( 3 \right)$ or $P\left( a \right)$.
Hence, we have found the value of $P\left( a \right)$ using the remainder theorem to be equal to $168$.

Note: Although we can simply substitute $x=3$ in the given polynomial $P\left( x \right)$ to find out the value of $P\left( 3 \right)$, but since the question directs us to use the remainder theorem for the same, we have to use it. But we can check the value obtained by substituting $x=3$ in the given polynomial, since there are chances of calculation errors in the division.