Let `R_1` and `R_2` are the remainders when the polynomials `x^3+2x^2-5a x-7` and `x^3+a x^2-12 x+6` are divided by `x+1` and `x-2` respecti-vely. If `2R_1+R_2=6,` find the value of `adot`

Explanation

Given the polynomials `P(x) = x³+2x^2-5ax-7` and `P'(x)= x³+ax²-12x+6`
when the above polynomials are divided by `x-1` and `x + 2` resp.
then R₁ and R2 are remainders such that`2R1 + R₂ = 2`
By remainder theorem
`P(1) = x³ + 2x²-5ax-7=1³+2(1)^2-5a(1)-7=1+2-5a-7=-4-5a`
`P'(-2)=³+ax²-12x+6= (-2) ³+ a(-2)2-12(-2) +6=8+4a +24+6 =22 + 4a`As given`2P(1) +P'(-2) = 2`
`2(-4-5a)+22+ 4a =-8-10a +22+ 4a = 2`
`6a= 12=a = 2`