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RS Aggarwal Solutions Class 9 Mathematics Solutions for Factorization of Polynomials Exercise 3B in Chapter 3 - Factorization of Polynomials
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𝑥^8 − 1
Answer:
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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