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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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