Difference of Cubes Formula

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Define Difference of Cubes Formula

In algebra, the cube of a number ‘n’ is its third power, i.e., the result of multiplying ‘n’ three times collectively. The cube of a number is expressed by a superscript 3, e.g.

23 = 8 or (x + 1)³.

Or, the cube can also be expressed as a number multiplied by its square:

n³ = n × n² = n × n × n.

The cube function is the function x ↦ x³ (often signified y = x³) that maps a number to its cube. It’s an odd function, as

(- n)³ = - (n³).


Difference of Cubes Formula 

The difference of cubes formula in algebra is used to calculate the value of the algebraic expression (a³ - b³). In simplistic words, it is applied to equate the difference of two cube values. Therefore the Formula for Difference of Cubes in Algebra is given as:

a³ - b³ = (a - b) (a² + ab + b²)


Difference of Two Cubes Formula

An expression that occurs in the difference of two cubes usually is very hard to spot. The difference between the two cubes is equal to the difference of their cube roots, which contains the cube roots' squares and the opposite of the cube roots' product.

To see what distribution results in the difference of two cubes formula, we try to see if the distribution has a binomial, 

(a - b), 

which is the difference between two terms

(a² + ab + b²)

which has the opposite of their product and the squares of the two terms . Therefore the formula for the difference of two cubes is -

a³ - b³  = (a - b) (a² + ab + b²)


Factoring Cubes Formula

We always discuss the sum of two cubes and the difference of two cubes side-by-side. The idea is that they are related to formation. The only solution is to remember the patterns involved in the formulas. 

Lets say -

Factoring  x³ - 8,

This is equivalent to x³ - 2³. As the - sign is in the middle, it transpires into a difference of cubes. To do the factoring, so plugging x and 2 into the difference-of-cubes formula. Doing so, we get:

x³ - 8 = x³ - 2³

= (x - 2)(x² + 2x + 2²)

= (x - 2)(x² + 2x + 4)


Conclusion 

The most popular perfect cubes are those whose roots are not decimals but are integers. To factor the two perfect cubes differences, remember that the difference of two perfect cubes equals the variance of their cube roots calculated by the product of their cube roots and the sum of their squares.

FAQ (Frequently Asked Questions)

Q1. What is the Factoring Cubes Formula?

Answer : The difference between the two cube formulas is in the area of the minus sign: For the difference of cubes, the - sign operates in the linear factor, a - b; for the sum of cubes, the - sign works in the quadratic factor, a²- ab + b².

Q2. What Steps are Needed for Factoring a Sum of Cubes?

Answer : To factor the sum of two cubes, determine if the two terms have the greatest common factor or GCF. Next, rewrite the first problem as a difference of two perfect cubes, then use those three pieces to write the final answer.

Q3. Explain How an Expression is a Sum of Cubes?

Answer : An expression requires two criteria to be factored as a sum of cubes. First, each term must be a cube. Secondly, each term should have the same sign, usually both positive.