Factoring Quadratic Expressions

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Factoring

Factoring is rewriting an expression as the product of its factors.  The greatest common factor (GCF) of an expression is the common factor with the greatest coefficient and the greatest exponent. It is possible to factor any expression containing terms with a GCF greater than 1.

Factoring is the reverse of multiplication.  To factor an expression means to write it as a product.  When factoring polynomials first look for the greatest common factors (GCF).

example:

Then proceed by considering the number of terms to be factored.

FACTORING TWO TERMS

If you are asked to factor a binomial there are only a few possible cases:

  1. the binomial is prime and cannot be factored
  2. the binomial contains a common factor (GCF)
  3. the binomial is the difference of two squares
  4. the binomial is the sum of two cubes
  5. the binomial is the difference of two cubes

Patterns to recognize:

Difference of Two Squares

Sum of Two Squares
Sum of Two Cubes
Difference of Two Cubes

 

examples:

 

prime
GCF

Difference of Two Squares
Sum of Two Cubes
Difference of Two Cubes

 

FACTORING THREE TERMS

If you need to factor a trinomial (three terms) of the form ax2 + bx + c, the possibilities to consider are: 

  1. Greatest common factor only.

  2. General trinomials-where leading coefficient is 1.

  3. Perfect square trinomial.

  4. Challenging trinomials-where leading coefficient is greater than 1.

  Patterns to recognize:

Perfect square trinomial
Perfect square trinomial

 

examples:

 


GCF only
General trinomial

Perfect square trinomial
Challenging trinomial
prime

 

Guess and Check Method:

To factor ax2 + bx + c, we look for binomials (__x + __)(__x + __) where products of numbers in the blanks are as follows:

  1. The numbers in the first blanks of each binomial have product a.

  2. The numbers in the last blanks of each binomial have product c.

  3. The outside product and the inside product have a sum ob b. 

This method works best when a = 1.  However, when a is a number greater than one, the box method could be best.

Box Method:

In the example shown below, you are to factor 8x2 + 10x – 3. By making a box it is easy to keep track of each term in the polynomial.  

First look for a factor common to all terms. There is none. 
Next look for two numbers whose product is 8.

1, 8

 or 

2, 4

Now look for numbers whose product is –3.
Since the last term in the polynomial is negative, the signs of the second terms must be opposite.

–1, 3

 or 

1, –3

Finally, make a box like the one shown below and begin to check the middle term by multiplying along the diagonals of the box.  Notice our first choice produced the desired middle term of +10x.  If this did not occur on the first attempt, you simply try your other factor choices or switch your choices in the boxes.  Notice if you multiply down the columns of the box your first term of 8x2 is in the first column and your constant term of –3 is in the second column.  

 

When every column checks with the polynomial that is to be factored, simply write the binomials from the rows of the box.

In this case, 8x2 + 10x – 3 = (4x – 1)(2x + 3)

FACTOR BY GROUPING

If you are asked to factor four terms try to factor by grouping two terms at a time.

example:

There is also the possibility that you have the difference of two squares again, in which case you can factor by grouping 3 terms and then 1 term as shown in the following example:

example: