Simplification of (a ± b)(a^2 ∓ ab + b^2)

It is very important for your child to know how to expand and how to simplify the binomial and trinomial expressions. Without knowing the simplifications they cannot solve the problems. On this page, you can find the methods to simplify (a ± b)(a^2 ∓ ab + b^2) expressions. Let us take some examples of simplification of (a ± b)(a^2 ∓ ab + b^2) and solve them in simple techniques.

Solving the problems in simple methods will help you to finish the exam time. So, we suggest the students practice the problems on Expansion of Powers of Binomials and Trinomials to become perfect in this chapter.

Simplification of (a ± b)(a^2 ∓ ab + b^2) Derivation

Simplification of (a + b)(a² – ab + b²):
(a + b)(a² – ab + b²) = a(a² – ab + b²) + b (a² – ab + b²)
(a + b)(a² – ab + b²) = a³ – a²b + ab² + ba² – ab² + b³
(a + b)(a² – ab + b²) = a³ + b³

Simplification of (a – b)(a² +ab + b²):
(a – b)(a² + ab + b²) = a(a² + ab + b²) – b (a² + ab + b²)
(a – b)(a² + ab + b²) = a³ + a²b + ab² – ba² – ab² – b³
(a – b)(a² + ab + b²) = a³ – b³

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Simplifying (a ± b)(a^2 ∓ ab + b^2) Examples

The simplification of (a ± b)(a^2 ∓ ab + b^2) is shown with some examples in the below section.

Example 1.
Simply the equation (3x + y) (9x² – 3xy + y²)
Solution:
Given that
(3x + y) (9x² – 3xy + y²)
(3x + y) {(3x)² – (3y)y + y²}
(3x)³ + y³ [ therefore (a + b)(a² – ab + b² = a³ + b³]
27x³ + y³
The simplification of (3x + y) (9x² – 3xy + y²) is 27x³ + y³

Example 2.
Simply the equation (x + 1/x)(x² + 1 + 1/x²)
Solution:
Given that
(x + 1/x)(x² + 1 + 1/x²)
(x + 1/x)( x² + x × 1/x + (1/x)³)
x³ – 1/x³ [ Since (a + b)(a² – ab + b² = a³ + b³]
The simplification of (x + 1/x)(x² + 1 + 1/x²) is x³ – 1/x³

Example 3.
Simply the equation (4x + y) (16x² – 4xy + y²)
Solution:
Given that
(4x + y) (16x² – 4xy + y²)
(4x + y) {(4x)² – (4y)y + y²}
(4x)³ + y³ [ since the formula of (a + b)(a² – ab + b² = a³ + b³]
64x³ + y³
The simplification of (4x + y) (16x² – 4xy + y²) is 64x³ + y³

Example 4.
Simply the equation (5x + y) (25x² – 5xy + y²)
Solution:
Given that
(5x + y) (25x² – 5xy + y²)
(5x + y) {(5x)² – (5y)y + y²}
(5x)³ + y³ [ therefore (a + b)(a² – ab + b² = a³ + b³]
125x³ + y³
The simplification of (5x + y) (25x² – 5xy + y²) is 125x³ + y³

Example 5.
Simply the equation (6x + y) (36x² – 6xy + y²)
Solution:
Given that
(6x + y) (36x² – 6xy + y²)
(6x + y) {(6x)² – (6y)y + y²}
(6x)³ + y³ [ therefore (a + b)(a² – ab + b² = a³ + b³]
216x³ + y³
The simplification of (6x + y) (36x² – 6xy + y²) is 216x³ + y³

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