Factorization of Expressions of the Form a$$^{3}$$ + b$$^{3}$$

Here we will learn the process of Factorization of Expressions of the Form a3 + b3.

We know that (a + b)3 = a3 + b3 + 3ab(a + b), and so

a3 + b3 = (a + b)3 – 3ab(a + b) = (a + b){(a + b)2 – 3ab}

Therefore, a3 + b3 = (a + b)(a2 – ab + b2)

Solved Examples on Factorization of Expressions of the Form a^3 + b^3

1. Factorize: x3 + 8y3

Solution:

Here, given expression = x3 + 8y3

= (x)3 + (2y)3

= (x + 2y){(x)2 – (x)(2y) + (2y)2}

= (x + 2y)(x2 – 2xy + 4y2).

2. Factorize: m6 + n6.

Solution:

Here, given expression = m6 + n6

= (m2)3 + (n2)3

= (m2 + n2){(m2)2 – m2 ∙ n2 + (n2)2}

= (m2 + n2)(m4 – m2n2 + n4)

3. Factorize: 1 + 125x3.

Solution:

Here, given expression = 1 + 125x3.

= 1^3 + (5x)3

= (1 + 5x){12 - 1 ∙ 5x + (5x)2}

=(1 + 5x)(1 - 5x + 25x2).

4.  Factorize: 8x3 + $$\frac{1}{x^{3}}$$

Solution:

Here, given expression = 8x3 + $$\frac{1}{x^{3}}$$.

= (2x)3 + ($$\frac{1}{x}$$)3

= (2x + $$\frac{1}{x}$$){(2x)2 - 2 ∙ x ∙ $$\frac{1}{x}$$ + ($$\frac{1}{x}$$)2}

= (2x + $$\frac{1}{x}$$)(4x2 - 2 + $$\frac{1}{x^{2}}$$).

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