· 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)
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)
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