# Find the value of a Given: Let A and B be the remainders when polynomials x3+2x2-5ax-7 and x3+ax2-12x+6 are divided by x+1 and x-2 respectively. If 2A+B=6 , find the value of a.

Let A and B be the remainders when polynomials x3+2x2-5ax-7 and x3+ax2-12x+6 are divided by x+1 and x-2 respectively. If 2A+B=6 , find the value of a.

is 'a' and 'A' different ?
Yes, a and A are different. When we divide the reminder we get A and B will be in terms of "a", so we need to solve for "a" given 2A+B = 6.

Toolbox:
• Remainder Theorem : Let $p(x)$ be an polynomial of degree greater than or equal to one.Let 'a' be any real number.If $p(x)$ is divisible by $(x-a)$ then the remainder is $p(a)$
Step 1:
$p(x)=x^3+2x^2-5ax-7$
Since $p(x)$ is divisible by $(x+1)$ the remainder is $p(-1)$
$p(-1)=-1+2+5a-7$
$\qquad\;\;=5a-6$
$\qquad\;\;=A$
Step 2:
$t(x)=x^3+ax^2-12x+6$
Since $t(x)$ is divisible by $(x-2)$ the remainder is $t(2)$
$t(2)=(2)^3+4a-24+6$
$\quad\;\;\;=4a-10$
$\quad\;\;\;=B$
Step 3:
Since $2A+B=6$
we know that $A=(5a-6),B=4a-10$
Therefore $2A+B=6\Rightarrow2(5a-6)+4a-10=6$
Solving for $'a'$
$10a-12+4a-10=6$
$14a-22=6$
$14a=28$
We get $a=2$
edited Jun 12, 2013