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# Factoring Cubic Equations: Techniques, Examples

November 2, 2023 written by Rida Mirza

Cubic equations, also known as third-degree equations, are a fascinating part of algebra.

In this article, we will discuss cubic equations, break down the factoring techniques, and provide solved examples.

## Cubic Equations

Cubic equations are polynomial equations of the form:

ax3 + bx2 + cx + d = 0

Here, a, b, c, and d are constants, and x is the variable.

## Factoring Techniques

### Grouping

Grouping is a useful technique to factor cubic equations. Start by grouping terms and looking for common factors.

### Example

Factor x3 + 6x2 + 11x + 6.

Group the terms.

x3 + 6x2 + 11x + 6

Factor out the common factors from each group.

= x2(x + 6) + 1(11x + 6)

Factor out the common factor.

= x2(x + 6) + 1(11x + 6)

= x2(x + 6) + 1(11x + 6)

### Factoring by Substitution

Another technique to factor cubic equations is substitution. You replace x with another variable to simplify the equation.

### Example

Factor x3 – 8.

Substitute (y = x3).

Now, the equation becomes (y – 8).

Factor (y – 8) as the difference of cubes.

= (y1/3)3 – 23

Now, substitute back y with x3.

= (x3)^{1/3} – 23

Simplify:

= x – 2

### Solved Example

Factor x3 – 27.

Substitute y = x3

Now, the equation becomes (y – 27).

Factor (y – 27) as the difference of cubes.

= (y1/3)3 – 33

Now, substitute back y with x3.

= (x3)^1/3 – 33

Simplify:

= x – 3

So, x3– 27 factors as x – 3x2 + 3x + 9.

## FAQs

### What is a cubic equation?

A cubic equation is a polynomial equation of the form ax3 + bx2 + cx + d = 0, where a,b, c, and (d) are constants, and (x) is the variable.

### How can I factor a cubic equation?

Cubic equations can be factored using techniques like grouping, substitution, or special factorizations, such as the difference of cubes.

### What is the difference of cubes?

The difference of cubes is a special factorization pattern, represented as a3 – b3 = (a – b)(a2 + ab + b2), which can be used to factor certain cubic equations.

### Are there any online tools to help factor cubic equations?

Yes, there are online cubic equation factoring calculators that can help you factor cubic equations quickly.

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