Question

The polynomials $a{x}^3-3x^2+4$ & $2x^3-5x+a$ when $(x-2)$ leaves remainders $p$ and $q$ & if $p-2q=4$ find $a$.

• A. $a=2$
• B. $a=4$
• C. $a=6$
• D. both a and b

Answer 1

The correct option is B $a=4$

It is given that a polynomial $a{x}^3-3x^2+4$ when divided by $(x-2)$ leaves the remainder $p$. Let us substitute $x=2$ in $a{x}^3-3x^2+4$ and equate it to $p$ as follows:

$a(2{)}^3-3(2{)}^2+4=p\Rightarrow (a\times 8)-(3\times 4)+4=p\Rightarrow 8a-12+4=p\Rightarrow 8a-8=p....\left(1\right)$

Similarly, given a polynomial $2x^3-5x+a$ when divided by $(x-2)$ leaves the remainder $q$. Let us substitute $x=2$ in $2x^3-5x+a$ and equate it to $q$ as follows:

$2(2{)}^3-(5\times 2)+a=q\Rightarrow (2\times 8)-10+a=q\Rightarrow 16-10+a=q\Rightarrow 6+a=q....(2)$

It is also given that $p-2q=4$, we, now substitute the values of $p$ and $q$ from equations 1 and 2 as shown below:

$p-2q=4\Rightarrow 8a-8-2(6+a)=4\Rightarrow 8a-8-12-2a=4\Rightarrow 6a-20=4\Rightarrow 6a=4+20\Rightarrow 6a=24\Rightarrow a=\frac{24}{6}\Rightarrow a=4$

Hence, $a=4$.