Q. 63.8( 33 Votes )

# If (x^{2} – 1) is a factor of ax^{4} + bx^{3} + cx^{2} + dx + e, show that a + c + e = b + d = 0

Answer :

Let f(x) = ax^{4} + bx^{3} + cx^{2} + dx + e

By Factor Theorem, we know that,

If p(x) is a polynomial and a is any real number, then g(x) = (x– a) is a factor of p(x), if p(a) = 0 and vice versa.

Also we can write, (x^{2} – 1) = (x + 1)(x – 1)

Since (x^{2} – 1) is a factor of f(x), this means (x + 1) and (x – 1) both are factors of f(x).

So, if (x – 1) is a factor of f(x)

⇒ f(1) = 0

⇒ a(1)^{4} + b(1)^{3} + c(1)^{2} + d(1) + e = 0

⇒ a + b + c + d + e = 0 ----- (A)

Also as (x + 1) is also a factor,

⇒ f(–1) = 0

⇒ a(–1)^{4} + b(–1)^{3} + c(–1)^{2} + d(–1) + e = 0

⇒ a – b + c – d + e = 0

⇒ a + c + e = b + d ---- (B)

On solving equations (A) and (B), we get,

a + c + e = b + d = 0

Rate this question :

If x^{2} – x – 6 and x^{2} + 3x – 18 have a common factor (x – a) then find the value of a.

Determine which of the following polynomials has (x + 1) as a factor.

x^{4} + 2x^{3} + 2x^{2} + x + 1

Use the Factor Theorem to determine whether g(x) is factor of f(x) in the following cases:

f(x) = x^{3} + 3x^{2} + 3x + 1, g(x) = x + 1

Factorise the following using appropriate identities.

AP- Mathematics

Factorize

x^{3} + 13x^{2} + 32x + 20

Factorize

x^{3} – 2x^{2} – x + 2