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Factoring Cubes is Not as Difficult as You Thought

Omkar Phatak Apr 21, 2019
If you are looking for tips on factoring cubes or cubic polynomials, you have landed on the right page. Along with a formula, you will find some illustrated examples.
Solving any kind of algebraic problem, requires the right technique, an attention to detail, and step-by-step calculation. The part of mathematics which always interested me, was algebra and it was fun solving polynomial problems.

Formulas

Algebraic formulas are keys that make it possible to unlock algebraic riddles. We suggest that you make a chart of all the algebraic formulas, which you need to use and stick it in front of your work desk.
They help you in simplifying any expression quite easily. In case of cubic polynomials, there are a couple of formulas, which can help you in factoring them quite easily. You need to remember the following four formulas:

a3 + b3 = (a +b) (a2 - ab + b2)

a3 - b3 = (a - b) (a2 + ab + b2)

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a - b)3 = a3 + 3a2b - 3ab2 + b3
If you observe these formulas, you will find a pattern in them, which makes them easy to be remembered and used. Solving algebraic equations is all about recognizing patterns, by grouping like terms together and using formulas like the ones listed before, to simplify them. All it takes is some keen observation and grouping of terms in the right order.

Solved Examples

The best way of mastering the solution of algebraic expressions like these is to practice solving factoring problems.

Factorize: x3 - 8

Solution: x3 - 8 = x3 - 23 = (x - 2) (x2 + 2x + 4)

Factorize: x3 + 343

Solution: x3 + 343 = x3 + 73 = (x + 7)(x2 - 7x + 49)
Factorize: x6 - 35x3 + 216

Solution: (x6 - 35x3 + 216) = (x6 - 8x3 - 27x3 + 216) = x3 (x3 - 8) - 27(x3 - 8) = (x3 - 8)(x3 - 27) = (x3 - 23)(x3 - 33) = (x - 2)(x2 + 2x + 4)(x - 3)(x2 + 3x + 9)

Factorize: m3 + 64

Solution: m3 + 64 = m3 + 43 = (m + 4)(m2 - 4m + 16)
As you can see through the examples, the trick is to take out the common factors and use the formulas listed before. Just make sure that you write the expression clearly at every stage and take care of the signs when taking the common factors out.
All that you need to do is bring the terms together which can enable the usage of the cubic polynomial formulas listed before. When it comes to algebra, you need to understand that you need to work smart, instead of just working hard.
The key to get better at solving algebraic problems is to practice solving as many as you can, on your own. That's the way to get better at mathematics and there is no other.