Question

When a polynomial f(x) is divided by (x^2-5) and the quotient is x^2-2x-3 and remainsre is zero. Find the polynomial and all its zeroes.

Answer 1

f(x) = ?

g(x) = x2 - 5

q(x) = x2 - 2x - 3

r(x) = 0

By division algorithm for polynomials, we have

f(x) = q(x) . g(x) + r(x)

f(x) = (x2 - 5)(x2 - 2x - 3) + 0

f(x) = x4 - 2x3 - 3x2 - 5x2 + 10x + 15

f(x) = x­4 - 2x3 - 8x2 + 10x + 15

So, the required polynomial is f(x) = x­4 - 2x3 - 8x2 + 10x + 15

Now,

q(x) and g(x) will be factors of f(x)

x2 - 5 = 0 and x2 - 2x - 3 = 0

x2 - (√5)2 = 0 and x2 + x - 3x - 3 = 0

(x - √5)(x + √5) = 0 and (x + 1)(x - 3) = 0

x = √5, x = -√5, x = -1 and x = 3

So, the zeroes are α = √5, β = -√5, γ = -1 and δ = 3