Factoring Polynomials

Rules:

·         a^2 - b^2 = (a + b) (a - b)

·          a^3 - b^3 = (a - b) (a^2 + ab + b^2)

·          a^3 + b^3 = (a + b) (a^2 - ab + b^2)

Problem:

t^21 + 1.

The only rule which looks like it might work is the third one above.
If we're going to apply it, we need to be able to express t^21 as the
cube of something (a in the rule as I've given it), and 1 as the cube
of something (b in the rule).  Can we do this?  Well, fortunately the
exponent 21 is a multiple of 3, namely 3 * 7, so by the rules of
exponents

 (t^7)^3 = t^(7 * 3) = t^21

So we can write t^21 = (t^7)^3 and t^7 can fit in the place of a in
the factoring rule.

 1 is no problem, since 1 = 1^3

So we have

 t^21 + 1 = (t^7)^3 + 1^3 = (t^7 + 1) ((t^7)^2 - t^7 + 1)

Applying the rule with a = t^7 and b = 1.

This can be simplified to

 (t^7 + 1) (t^14 - t^7 + 1)

A few problems and examples:

25y^2 - 144 = 0

As you've probably guessed, you're going to solve this by factoring.

This time the rule that we can hope to apply is the first rule, so we
have to write

 25y^2 - 144

As a difference of two squares.

Fortunately, we have 25 = 5^2, so 25y^2 = (5y)^2; and 144 = 12^2.
So we can write

 25y^2 - 144 = (5y)^2 = 12^2

and we can apply the first rule with a = 25y and b = 12 to get

 25y^2 - 144 = (5y + 12) (5y -12)

and we set this equal to zero:

 (5y + 12) (5y - 12) = 0

The only way to get a product to come out zero is for one of the
factors to be zero, so to get the equation we have to have either

 5y + 12 = 0

or

 5y - 12 = 0

So that y = -12/5 and y = 12/15 are the solutions.

·         http://mss.math.vanderbilt.edu/~pscrooke/MSS/factorpoly.htm

·         http://icm.mcs.kent.edu/research/facdemo.html

·         http://www.sosmath.com/algebra/factor/fac02/fac02.html

·         http://library.thinkquest.org/20991/alg2/polyf.html#factor

·         http://cap.epsb.ca/math14_Jim/math9/strand2/2210.htm

·         http://math.usask.ca/readin/examples/par_eg6.html

·         http://homepage.mac.com/shelleywalsh/Math%20Articles/Factoring.html

Example Graph:

3t^2-2t-1=0

 

Some history about factoring polynomials:

 

·        What is Factoring?
Recall the distributive law (BR-9) from the Basic Rules of Algebra section. This is generally called "expanding" while doing this rule in reverse is called "factoring". The expressions to either expand or factor are usually more complex than a(b + c) = ab + ac, but the same procedure is followed (recall the "FOIL" method discussed in the Basic Rules of Algebra section).

Essentially, factoring is rewriting an expression as a product of 2 or more expressions. [Recall back to elementary mathematics where factoring a number, such as 15, would be writing it as the multiple of two other numbers, such as 15 = (5)(3) where both 5 and 3 are factors of 15.] While a polynomial is an expression involving powers of x (or any variable) that involves only addition and multiplication as the types of arithmetic operations used. Each term in a polynomial can be written as axj where a is a real number and j is a non-negative integer. There is more information about this in the Polynomials and Roots section.

Common Factors
One of the most basic ways to factor an expression is to "take out a common factor". If every term in an expression has several factors, and if every term has at least one factor that is the same, then that factor is called a common factor. If this is the case, then the common factor can be "taken out" of every term and multiplied by the whole remaining expression.

 

About the creator:

·        My name is Amy L. Rosales I am the creator of this web page. I am a junior here at Fayetteville High School. We were given this assignment to see what different areas we have covered in algebra 2. arosales@fayar.net