Math Review of Special Factoring Formulas and a General Review of Factoring

Overview

Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping.

Extending the Difference of Two Squares

The difference of two squares [(a + b)(a – b)] is a common pattern with binomials involving variables to the second power. However, the concept can also be applied to exponents higher than x². Any even power (such as x², x⁴, x⁶, and so on) can be factored into squares evenly. For example, 16x⁴ can be rewritten as (4x²)² and 49y⁶ can be rewritten as (7y³)². The expression 16x⁴ – 49y⁶ is then factored as (4x² – 7y³)(4x² + 7y³).

Sum of Two Cubes

The product of (a + b)(a² – ab + b²) can be evaluated using FOIL as a³ –a²b + ab² +a²b – ab² + b³. That simplifies to a³ + b³. Suppose that the binomial that needs to be factored is 27x³ + 8. That expression will factor as (3x + 2)(9x² – 6x + 4).

Figure 1: Factoring the sum of two cubes and the difference of two cubes.

Difference of Two Cubes

The product of (a – b)(a² + ab + b²) can also be evaluated and simplified to a³ – b³. The easiest way to remember the direction of the signs when factoring the sum or difference between two cubes is to use the acronym SOAP. The sign between the terms of the binomial factor is in the same direction in both the sum of the cubes and the (a + b) factor. (If the difference of cubes is the issue, the sign in a³ – b³ and a – b is negative.) The sign is opposite between the a² term and the ab term, such that if it is the sum of cubes the sign between a² and ab is negative, and if it is the difference in cubes, the sign between the a² and ab term is positive. The sign between the ab term and the constant is always positive.

Figure 2: Using the acronym SOAP to remember the direction of the signs.

Extending Factoring Strategies

The first step in factoring a polynomial is always to factor out anything that is common to every term in the polynomial. Suppose that the polynomial to be factored is 3x² + 6x + 9. The first step in factoring would be to remove the common factor of 3 from all the terms as 3(x² + 2x + 3). Next, check to see if it follows any of the special factoring forms. It can be factored by grouping or another method.

Figure 3: Following the general steps to factor a polynomial.

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