Question

if zeroes of the polynomial p(x)=3x³-5x²-11x-3 are α,β and  γ, then find the following (a) α+β+γ , (b) αβ+βγ+γα , (c) αβγ

Answer 1

Comparing the given polynomial with ax3 + bx2 + cx + d, we get

a = 3, b = – 5, c = –11, d = – 3. Further

p(3) = 3 × 33 – (5 × 32) – (11 × 3) – 3 = 81 – 45 – 33 – 3 = 0,

p(–1) = 3 × (–1)3 – 5 × (–1)2 – 11 × (–1) – 3 = –3 – 5 + 11 – 3 = 0,

p(-1/3) = 3* (-1/3)3 - 5*(-1/3)2 - 11*(-1/3) - 3,

= -1/9-5/9+11/3-3 = 0

Therefore, 3, -1 and -1/3 are the zeroes of 3x3 - 5x2 -11x -3

So, we take α = 3, β = -1, γ = -1/3,

Now,

α + β + γ = 5/3 = -(-5)/3 = -b/a,

αβ + βγ + γα = -11/3 = c/a,

αβγ = 1 = -(-3)/3 = -d/a