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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