Question

if x^2-1 is a factor of ax^3+bx^2+cx+d, show that a+c=0.

Answer 1

Given:-

• f(x) = ax³ + bx² + cx + d

• (x² - 1) is a factor of f(x)

To prove:-

• a + c = 0

Solution:-

We have, x² - 1 = 0

So, x² = 1

→ x = ± 1

★ Put x = -1 :-

→ a(-1)³ + b(-1)² + c(-1) + d = 0

→ - a + b - c + d = 0 ---(1)

★ Now, Put x = 1 :-

→ a(1)³ + b(1)² + c(1) + d = 0

→ a + b + c + d = 0 ---(2)

★ Substract eq(2) from eq(1) :-

Then we get,

→ - 2a - 2c = 0

→ - 2(a + c) = 0

→ a + c = 0

★ Hence, SolvEd!!!