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 + \(\frac{1}{x^{2}}\)).










9th Grade Math

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