FACTORING

QUADRATICS

WHAT TO LOOK FOR

GCF SHORT CUTS

GENERIC RECTANGLE/

DIAMONDLook for only two

terms.Look for only two

terms.Look for three terms.

Look for a coefficient in front of x.

Look for two perfect squares

being subtracted.

The first term must be squared.

Look for the variable to be cubed

The beginning & end terms are

perfect squares and the middle

term is doubled.

The coefficient of x is the sum of two

numbers and the constant is the product

of the same two numbers.

VOCABULARY Difference of squares - A special polynomial that can be factored as the

product of the sum and difference of two terms. Factor - Where two or more algebraic expressions are multiplied together,

each of the expressions is a factor of the product. Factored completely - A polynomial is factored completely if none of the

resulting factors can be factored further. Generic rectangle - An organizational device used for multiplying and

factoring polynomials. Greatest common factor - for a polynomial, the greatest common monomial

factor of its terms. Perfect square trinomials - Trinomials of the form are

known as perfect square trinomials as Polynomial - the sum or difference of two or more monomials. Quadratic - A polynomial is quadratic if the largest exponent in the

polynomial is two (that is, the polynomial has degree 2). Term - Each part of the expression separated by addition or subtraction

signs. Variable - A variable is a symbol used in a mathematical sentence to

represent a number.

€

x 2 + 2ax + a2

€

(x + a)2

GCF - TWO TERMS A big clue that you need to use the

GCF is when there are only two terms

Examples

€

4x −16 3x 2 +12x

GCF - coefficient in front of x (a > 1) Another clue that you will need to

use the GCF is when the coefficient in front of x is greater than 1.

Examples

€

2x 2 +10x + 24 3x 2 −15x + 9

GCF - X is cubed A third clue that you should use the

GCF to factor is when the variable is cubed.

Examples

€

x 3 −12x 2 + 20x 4x 3 + 8x 2 −16

DIFFERENCE OF SQUARES (Short Cut)Here’s what to look for with difference of squares:a. There are only two termsb. Both terms are perfect squares. c. The terms must be subtracted!

Once you determine an expression is a difference of

squares it’s very simple to factor. For example:

€

x 2 −16 = (x − 4)(x + 4)

x 2 − 36 = (x − 6)(x + 6)

PERFECT SQUARE TRINOMIAL (Short cut)

DESCRIPTION If the first and third terms are squares, take their

squareroot, multiply them together and then multiply by

2. Ifyour solution is the quadratic's middle term, then

you'vegot a perfect square trinomial.EXAMPLES

€

x2 − 12x + 36

€

x2 + 16x + 64

€

x 2 +14x + 49

FACTORING COMPLETELY A polynomial is factored completely

if none of the resulting factors can be factored further.

Examples This polynomial is factored completely:

€

6x 2 − 30x + 36 = 6(x 2 − 5x + 6) = 6(x − 3)(x − 2)

GENERIC RECTANGLE & DIAMOND - SIMPLE After the GCF and the factoring short cuts, the next thing

you should look for when factoring is to see if you can use the generic rectangle and diamond method.

Diamond Problems can be used to help factor easier quadratics like x2 + 6x + 8.

2 4

6

8

x2

4x

2x 8

x2

4x

2x 8

x + 4

x

+

2

(x + 4)(x + 2)

GENERIC RECTANGLE & DIAMOND - COMPLEX We can modify the diamond method slightly to factor

problems that are a little different in that they no longer have a “1” as the coefficient of x2. For example, factor:

2x + 7x + 3

2

6

7

? ?

6

7

6 1

2x 6x

1x 3

2

2x 6x

1x 3

2

2x

+

1

x + 3

multiply

(2x + 1)(x + 3)

REFERENCESSallee, T., Kysh, J., Kasimatis, E.,(2002). CPM Algebra 1.Sacramento, CAhttp://www.saab.org/mathdrills/factor.cgi - factoring practicehttp://www.regentsprep.org/Regents/math/ALGEBRA/AV6/PracFact1.htm -

DOShttp://www.purplemathhttp://www.mathvids.com/lesson/mathhelp/790-factoring-polynomials-using-gcf

http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_Factoring.xml

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