Question

It is known that $p\left(x\right)=x^2-4x+a$ and $q\left(x\right)=b{x}^2+3x+1$ leave the same remainder when divided by $x-2$. Which of the following relationships between a and b is true?

• A. a + b = 4
• B. a + 4b = -1
• C. a - 4b = 1
• D. a - 4b = -1

Answer 1

The correct option is D a - 4b = -1

From remainder theorem, we know
Remainder when $p\left(x\right)$ is divided with $x-2$ is $p\left(2\right)$.
$\therefore$ Remainder = p(2)
= $2^2-4\left(2\right)+a$
$\Rightarrow 4-8+a$
$\Rightarrow a-4$

Similarly,
Remainder when $q\left(x\right)$ is divided with $x-2$ is $q\left(2\right)$.
$\therefore$ Remainder = q(2)
$=b\left(2^2\right)+3\left(2\right)+1$
$\Rightarrow 4b-6+1$
$\Rightarrow 4b-5$

Given, these two remainders are equal
$\Rightarrow a-4=4b-5$
$\Rightarrow a-4b=4-5$
$\Rightarrow a-4b=-1$