# RS Aggarwal Solutions Class 9 Mathematics Solutions for Factorization of Polynomials Exercise 3B in Chapter 3 - Factorization of Polynomials

Question 36 Factorization of Polynomials Exercise 3B

𝑥^8 − 1

𝑥^8 − 1

We can further write it as

= (𝑥^4)^2 − 1^2

Based on the equation 𝑎^2 − 𝑏^2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

We get,

= (𝑥^4 + 1)(𝑥^4 − 1)

(𝑥^4 − 1) Can also be written using the formula

= (𝑥^4 + 1)((𝑥^2)^2 − 1^2)

So we get

= (𝑥^4 + 1)(𝑥^2 + 1)(𝑥^2 − 1)

By expanding the terms using the formula

We get,

= [(𝑥^2)^2 + 1^2 + 2𝑥^2 − 2𝑥^2](𝑥^2 + 1)(𝑥 + 1)(𝑥 − 1)

On further simplification

= [((𝑥^2)^2 + 1^2 + 2𝑥^2) − 2𝑥^2](𝑥^2 + 1)(𝑥 + 1)(𝑥 − 1)

So we get

= [(𝑥^2 + 1) − (√2𝑥)^2] (𝑥^2 + 1)(𝑥 + 1)(𝑥 − 1)

Using the formula we get,

= (𝑥^2 + 1 + √2𝑥) (𝑥^2 + 1 - √2𝑥) (𝑥^2 + 1)(𝑥 + 1)(𝑥 − 1)

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