# Factoring Cubic Equations: Techniques, Examples

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:

ax^{3} + bx^{2} + 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 x^{3} + 6x^{2} + 11x + 6.

Group the terms.

x^{3} + 6x^{2} + 11x + 6

Factor out the common factors from each group.

= x^{2}(x + 6) + 1(11x + 6)

Factor out the common factor.

= x^{2}(x + 6) + 1(11x + 6)

= x^{2}(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 x^{3} – 8.

Substitute (y = x^{3}).

Now, the equation becomes (y – 8).

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

= (y^{1/3})^{3} – 2^{3}

Now, substitute back y with x^{3}.

= (x^{3})^{1/3} – 2^{3}

Simplify:

= x – 2

**Solved Example**

Factor x^{3} – 27.

Substitute y = x^{3}

Now, the equation becomes (y – 27).

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

= (y^{1/3})^{3} – 3^{3}

Now, substitute back y with x^{3}.

= (x^{3})^1/3 – 3^{3}

Simplify:

= x – 3

So, x^{3}– 27 factors as x – 3x^{2} + 3x + 9.

**FAQs**

### What is a cubic equation?

A cubic equation is a polynomial equation of the form ax^{3} + bx^{2} + 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 a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2}), 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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