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:

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.

= (y^1/3)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.

= (y^1/3)³ – 3³

Now, substitute back y with x³.

= (x³)^1/3 – 3³

Simplify:

= x – 3

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

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