Question

The polynomial $p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$ when divided by $x+1$ leaves the remainder $19$.
Find the value of $a$. Also find the remainder when $p\left(x\right)$ is divided by $x+2$.

• A. $a=5;62$
• B. $a=4;62$
• C. $a=5;60$
• D. $a=4;60$

Answer 1

The correct option is A $a=5;62$

The given polynomial is $p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$
Given that, the polynomial $p\left(x\right)$ when divided by $(x+1)$ leaves remainder $19$
Therefore, $p(-1)=19$ (By Remainder theorem)
$=>(-1{)}^4-2\times (-1{)}^3+3(-1{)}^2-(-1)a+3a-7=19$
$=>1+2+3+a+3a-7=19$
$=>4a-1=19$
$=>4a=20$
$=>a=5$
The value of $a$ is $5$
Now,
$p\left(x\right)=x^4-2x^3+3x^2-5x+3\times 5-7$
$=x^4-2x^3+3x^2-5x+15-7$
$=x^4-2x^3+3x^2-5x+8$
Remainder when the polynomial is divided by $(x+2)$
$=p(-2)$ (By Remainder Theorem)
$={-2}^4-2(-2{)}^3+3(-2{)}^2-5(-2)+8$
$=16+16+12+10+8$
$=62$
Thus, the remainder of the polynomial $p\left(x\right)$ when divided by $(x+2)$ is $62$