If `x-2` is a factor of each of the following two polynomials, find the values of `a` in each case. (i) `x^3-2a x^2+a x-1` (ii) `x^5-3x^4-a x^3+3a x^2+2a x+4`

Explanation

1409724 (i) Let `f(x)=x^3-2ax^2+ax-1` be the given polynomials.
By factor theorem, if `(x−2)` is a factor of `f(x)`, then `f(2)=0`
`∴f(2)=0`
`⇒(2)^3−2a(2)^2+a(2)−1=0`
`⇒8−8a+2a−1=0`
`⇒7−6a=0`
`⇒6a=7`
`∴ a=7/6`

(ii) Let `f(x)=x^5−3x^4−ax^3+3ax^2+2ax+4` be the given polynomial.
By factor theorem, if `(x−2)` is a factor of `f(x)`, then `f(2)=0`
`∴f(2)=0`
`⇒(2)^5−3(2)^4−a(2)^3+3a(2)^2+2a(2)+4=0`
`⇒32−48−8a+12a+4a+4=0`
`⇒−12+8a=0`
`⇒8a=12`
`⇒a=12/8`
`∴a=3/2`