Let A and B be the remainders when polynomials x^{3}+2x^{2}-5ax-7 and x^{3}+ax^{2}-12x+6 are divided by x+1 and x-2 respectively. If 2A+B=6 , find the value of a.

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^{3}+2x^{2}-5ax-7 and x^{3}+ax^{2}-12x+6 are divided by x+1 and x-2 respectively. If 2A+B=6 , find the value of a.

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- 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 $a=2$

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