Question

Let p(x) be a quadratic polynomial such that $p\left(0\right)=1$. If p(x) leaves remainder $4$ when divided by $x-1$ and it leaves remainder $6$ when divided by $x+1$; then which one is correct?

• A. $p\left(2\right)=11$
• B. $p(-2)=19$
• C. $p(-2)=11$
• D. $p\left(2\right)=19$

Answer 1

Answer: (B)

$p(-2)=19$

$p\left(x\right)=a{x}^2+bx+c$

$p\left(0\right)=1=c=1$

$\begin{array}{c}p\left(1\right)=4\\ p(-1)=6\end{array}]$

$\begin{array}{c}a+b+c=4\\ a-b+c=6\end{array}]$
subtracting the above equations, we get $2b=-2\Rightarrow b=-1$
adding the above two equations and putting values of $c,$ we get
$2(a+c)=10$
$\Rightarrow a+c=5$
$\Rightarrow a+1=5$
$\Rightarrow a=4$

putting the values in the specified general equation,

$p\left(x\right)=4x^2-x+1$

$p(-2)=16+2+1=19$.