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The polynomials a x^3+3x^2-3 and 2x^3-5x...

The polynomials `a x^3+3x^2-3` and `2x^3-5x+a` when divided by `(x-4)` leave the remainders `R_1a n dR_2` respectively. Find the values of a in each of the following cases, if `R_1=R_2` (ii) `R_1+R_2=0` `2R_1-R_2=0`

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"Solution
By Remainder Theorem,
`R₁ = a(4)^3 +3(4)^2-3 = 64a +45...(i)`
`R₂ = 2(4)^3-5(4)+ a = 128-20+ a = 108 + a...(ii)`
Given: `2R1 R₂ = 0`
`:.2(64a + 45) - (108+ a) = 0` (from `(i)` and `(ii)`)
`⇒128a+90-108-a = 0`
⇒ `127a = 18
:.a =18/127`"
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