Question

If the polynomial 6x^4 + 8x^3 – 5x^2 + ax + b is exactly divisible by the polynomial 2x^2 – 5, then find the values of a and b.

Answer 1

From the above question we got,

the above polynomial is exactly divisible by 2x^2 -5..it imples that the remainder is 0

so, we can write it as

6x^4 + 8x^3 - 5x^2 + ax + b = (2x^2 - 5)× f(x)

where f(x) is a polynomial of 2nd degree

simplifying this furthur..we got at

6x^4 + 8x^3 - 5x^2 + ax + b = (√2x - √5)(√2x +√5)× f(x)

now R.H.S acqiure 0 when x =√(5/2) and x = -√(5/2)

by substituting x =√(5/2)

6(25/4) +8(5√5/2√2) - 5(5/2) + a(√5/√2) +b =0

25 + 20(√5/√2) + a(√5/√2) +b =0 -----(1)

by substituting x = -√(5/2)

6(25/4) - 8(5√5/2√2) - 5(5/2) - a(√5/√2) +b =0

25 - 20(√5/√2) - a(√5/√2) +b =0 -----(2)

(1) + (2)

50 +2b =0
b = - 25


(1) - (2)

40(√5/√2) + 2a(√5/√2) =0
a = -20