Question

Which long division problem can be used to prove the formula for factoring the difference of two perfect cubes?

'+ ab +6²
a+b) a² - ab +6²
a+b a² +0a²b+Oab² – 83
a-ba? +Da+b+Dab2 – 67

Answer 1

a³-b³ = (a-b)(a²+ab+b²)

Explanation:

let the two perfect cubes be a³ and b³. Factring the difference of these two perfect cubes we have;

a³ - b³

First we need to factorize (a-b)³

(a-b)³ = (a-b) (a-b)²

(a-b)³ = (a-b)(a²-2ab+b²)

(a-b)³ = a³-2a²b+ab²-a²b+2ab²-b³

(a-b)³ = a³-b³-2a²b-a²b+ab²+2ab²

(a-b)³ = a³-b³ - 3a²b+3ab²

(a-b)³ = (a³-b³) -3ab(a-b)

Then we will make a³-b³ the subject of the formula from the resultinh equation;

a³-b³ = (a-b)³+ 3ab(a-b)

a³-b³ = a-b{(a-b)²+3ab}

a³-b³ = a-b{a²+b²-2ab+3ab}

a³-b³ = (a-b)(a²+b²+ab)

a³-b³ = (a-b)(a²+ab+b²)

The long division problem that can be used is (a-b)(a²+ab+b²)