In this publication, we will consider one of the abbreviated multiplication formulas – the sum of cubes, with the help of which the expression is factored. We will also analyze examples of solving problems to consolidate the presented material.
Sum of cubes formula
The sum of cubes of numbers/expressions is equal to the product of their sum by the incomplete square of their difference.
a3 + b3 = (a + b)(a2 – ab + b2)
The full square of the difference looks like this: (a – b)2 = A2 – 2ab + b2. In our case, in the second bracket, instead of a double product, there is a single one, so the expression is called incomplete.
The formula is valid from right to left:
(a + b)(a2 – ab + b2) = the3 + b3
Note: a3 + b3 ≠ (a + b)3
Proof of the formula
You can verify the correctness of the expression by simply multiplying the brackets, observing the rules of arithmetic when opening them. Let’s do this:
(a + b)(a2 – ab + b2) = the3 – and2b+ab2 + a2b–ab2 + b3 = A3 + b3.
Examples of tasks
Task 1
Factor the expression: 63 + (4x)3.
Solution
63 + (4x)3 = (6 + 4x)(62 – 6 ⋅ 4x + (4x)2) = (6 + 4x)(36 – 24x + 16x2)
Task 2
Expand the expression into a product of factors: (7x)3 + (3y2)3.
Solution
(7x)3 + (3y2)3 = (7x + 3y2)((7x)2 – 7x ⋅ 3y2 + (3y)2) = (7x + 3y2)(49x2 – 21xy2 + 9y2)
Task 3
Represent the expression 64x3 + 125 as a sum of cubes and factor it.
Solution
64x3 + 125 = (4x)3 + 53 = (4x + 5)((4x)2 – 4x ⋅ 5 + 52) = (4x + 5)(16x2 – 20x + 25)