In each of the following two polynomials, find the value of `a ,` if `x+a` is a factor. (i) `x^3+a x^2-2x+a+4` (ii)`x^4-a^2x^2+3x-a`

Explanation

(i) Let `f(x)=x^3+ax^2−2x+a+4` be the given polynomial.
By factor theorem, if `(x+a)` is a factor of `f(x)`, then `f(−a)=0`
`∴f(−a)=0`
`⇒(−a)^3+a(−a)^2−2(−a)+a+4=0`
`⇒−a^3+a^3+2a+a+4=0`
`⇒3a=−4`
`⇒a=−4/3`
`therefore` The value of `a=-4/3`.

(ii) Let `f(x)=x^4−a^2x^2+3x−a` be the given polynomial.
By factor theorem, If `x+a` is a factor of `f(x)` then `f(−a)=0`
`⇒(−a)^4−a^2(−a)^2+3(−a)−a=0`
`⇒a^4−a^4−3a−a=0`
`⇒−4a=0`
`⇒a=0`
`therefore` The value of `a=0`.