Monomial Factors of Polynomials
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Key Questions

Answer:
The 'normal' way of finding the GCF of two polynomials is to factor both of them completely, then pick out the common factors and multiply them together.
Alternatively you can use division.
Explanation:
Normal Method  Factor both polynomials first
For example:
Given
#f(x) = x^2+6x+9# and#g(x) = x^2+x6# ,you can factor
#f(x) = (x+3)^2# and#g(x) = (x+3)(x2)# hence the GCF is
#(x+3)# Backup Method  Using division of polynomials
Given:
#f(x) = x^4+2x^3+4x^2+3x+2#
and:#g(x) = x^4+3x^3+6x^2+5x+3# If
#f(x)# and#g(x)# have a common polynomial factor#p(x)# then if we divide#f(x)# by#g(x)# (or vice versa), the remainder must be a multiple of#p(x)# .For example,
#g(x) = f(x) + (x^3+2x^2+2x+1)# Let
#h(x) = x^3+2x^2+2x+1# If we now divide
#f(x)# by#h(x)# then any remainder will also be divisible by#p(x)# :#x^4+2x^3+4x^2+3x+2# #= (x^3+2x^2+2x+1)x+2x^2+2x+2# #=h(x)*x + 2(x^2+x+1)# Next try dividing
#f(x)# by#x^2+x+1# ...#x^4+2x^3+4x^2+3x+2 = (x^2+x+2)(x^2+x+1)# This time there is no remainder, so
#p(x) = x^2+x+1# is our GCFFor a more complex example, see http://socratic.org/questions/ifyouaretoldthatx73x5x44x24x40hasatleastonerepeatedroothow

You can check your factoring by multiplying them all out to see if you get the original expression. If you do, your factoring is correct; otherwise, you might want to try again.
I hope that this was helpful.

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Questions
Polynomials and Factoring

1Polynomials in Standard Form

2Addition and Subtraction of Polynomials

3Multiplication of Monomials by Polynomials

4Multiplication of Polynomials by Binomials

5Special Products of Polynomials

6Monomial Factors of Polynomials

7Zero Product Principle

8Factorization of Quadratic Expressions

9Factor Polynomials Using Special Products

10Factoring by Grouping

11Factoring Completely

12Probability of Compound Events