Return to Free Math Homework Help   >>   Return to Algebra Main Page   >>   Return to Functions Main Page   >>   Return to Polynomials Main Page   >>   Return to Quadratic Equations Main Page   >>   Return to Factoring Methods Main Page

Free Homework Math Help: Special Factoring Formulas



        

Special Factoring Formulas

One of the tasks a student of algebra is expected to learn is pattern recognition. Linear and quadratic equations are recognized by their different patterns. This ability becomes even more important in higher math courses such as calculus. Some quadratic equations come in a recognizable form that can be factored more easily than with other methods. All that is required is that a few simple formulas or rules be learned. These formulas are the patterns. There are factoring formulas for both quadratic and cubic equations.

The special factoring formulas are as follows:

Perfect Square Trinomials         x2 ± 2xy + y2 = (x ± y)2
Difference of Two Squares         x2 - y2 = (x + y)(x - y)
Sum of Two Cubes         x3 + y3 = (x + y)(x2 - xy + y2)
Difference of Two Cubes         x3 - y3 = (x - y)(x2 + xy + y2)

Perfect Square Trinomials

In the Perfect Square Trinomial the sign of the middle coefficient will be the same sign in the factors. Example:

               x2 + 4x + 4       y2 = 4, y = 2
               x2 + 4x + 4 = (x+2)2

This example shows how the geometric square is related to the Perfect Square Trinomial rule in algebra.

Applying the Perfect Square Trinomial relies on pattern recognition

Another example:

               The formula: a2 - 2ab + b2
               4x2 - 12x + 9
               a2 = 4x2 a = 2x
               b2 = 9, b = -3       Since the middle coefficient is negative
               2ab = 22x-3 = -12x
               4x2 - 12x + 9 = (2x-3)2

Another visual example of the Perfect Square Trinomial.

The Perfect Square Trinomial is a rule used in algebra
The Perfect Square Trinomial is a formula for factoring
The Perfect Square Trinomial is used for factoring quadratic equations

Difference of Two Squares

In the Difference of Two Squares, the middle coefficient of the trinomial is missing. This is because it is cancelled out when multiplying and then adding factors. Example:

               Rule: a2 - b2
               25x2 - 9x2 = (5x - 3y)(5x + 3y)

The Difference of Squares is demonstrated visually with squares and rectangles from geometry.

The Difference of Squares makes factoring easy
The Difference of Squares is used to factor certain quadratic equations
The Difference of Squares is based on pattern recognition
The Difference of Squares can solve quadratic functions

Sum of Two Squares

It is important to note that the sum of two squares does not have real factors. However, it can be factored to complex conjugate factors. Example:

               4x2 + 9 = (2x + 3i)(2x - 3i)

Sum and Difference of Two Cubes

The sum and difference of cubes should not be confused with the sum and difference of squares. The quadratic factor in the rule for cubes is not a perfect square since there is no coefficient of two.

Sum of Cubes example

               Using this formula: a3 + b3 = (a+b)(a2 - ab + b2)
               Apply to: x3 + 27
               a = x
               b = 3
               x3 + 27 = (x+3)(x2 - 3x + 9)

Difference of Cubes example

               Using this formula: a3 - b3 = (a-b)(a2 + ab + b2)
               Apply to: 8x3 - 125
               a = 2x
               b = 5
               8x3 - 125 = (2x-5)(4x2 + 10x + 25)

The answers can be checked by using distribution. For quadratic Squares, FOIL can also be used to check answers.