Factorization of Polynomials | Factorization of Polynomials Exercise 3F

Question

8(𝑥 + 𝑦)^3 − 27(𝑥 − 𝑦)^3

Accepted Answer

8(𝑥 + 𝑦)^3 − 27(𝑥 − 𝑦)^3

We can write the given question as

= 2^3(𝑥 + 𝑦)^3 − 3^3(𝑥 − 𝑦)^3

According to the equation

𝑎^3 − 𝑏^3 = (𝑎 − 𝑏)(𝑎^2 + 𝑎𝑏 + 𝑏^2)

We get,

= (2(𝑥 + 𝑦) − 3(𝑥 − 𝑦))((2(𝑥 + 𝑦))^2+ 2(𝑥 + 𝑦)^3(𝑥 − 𝑦) + (3(𝑥 − 𝑦))^2)

According to the equation,

(𝑎 + 𝑏)^2 = 𝑎^2 + 2𝑎𝑏 + 𝑏^2

(𝑎 − 𝑏)^2 = 𝑎^2 − 2𝑎𝑏 + 𝑏^2

So we get,

= (2𝑥 + 2𝑦 − 3𝑥 + 3𝑦)((4(𝑥^2 + 2𝑥𝑦 + 𝑦^2) + 6(𝑥^2 − 𝑦^2) + (9(𝑥^2 − 2𝑥𝑦 + 𝑦^2))

= (−𝑥 + 5𝑦)(4𝑥^2 + 8𝑥𝑦 + 4𝑦^2 + 6𝑥^2 − 6𝑦^2 + 9𝑥^2 − 18𝑥𝑦 + 9𝑦^2)

= (−𝑥 + 5𝑦)(19𝑥^2 + 7𝑦^2 − 10𝑥𝑦)

Question 34