Question

The product of two expressions is x⁵ + x³ + x. If one of them is x² + x + 1, find the other.

Answer 1

We have, product of two expressions x⁵ + x³ + x and one is x² + x + 1.

Let the other expression be A.

Then, A × (x² + x + 1) = x⁵ + x³ + x

⇒ `A = (x^5 + x^3 + x)/(x^2 + x + 1)`

= `(x(x^4 + x^2 + 1))/(x^2 + x + 1)`

⇒ `A = (x(x^4 + 2x^2 - x^2 + 1))/(x^2 + x + 1)`

= `(x(x^4 + 2x^2 + 1 - x^2))/(x^2 + x + 1)`  ...[Adding and Subtracting x² in numerator term]

= `(x[(x^4 + 2x^2 + 1) - x^2])/(x^2 + x + 1)`

= `(x[(x^2 + 1)^2 - x^2])/(x^2 + x + 1)`

= `(x(x^2 + 1 + x)(x^2 + 1 - x))/(x^2 + x + 1)`   ...[Using the identity, a² – b² = (a + b)(a – b)]

= x(x² + 1 – x)

Hence, the other expression is x(x² – x + 1).