Algebra and Geometry Review – Exercise 06


QUESTION 1: Introduction to the GCF of two monomials

Find the greatest common factor of: 12x^2 and 8x^4

EXPLANATION:
The GCF of 12 and 8 is 4
The GCF of x^2 and x^4 is x^2
So, the GCF of 12x^2 and 8x^4 is 4x^2

ANSWER: 4x^2

QUESTION 2: Factoring a difference of squares in one variable

                Factor: u^2 – 4

EXPLANATION: Here is the factoring formula for the difference of squares
                A^2 – B^2 = (A + B)(A - B)
                We can use this for the current problem
                u^2 – 4 = u^2 – 2^2         Writing as A^2 – B^2, with A=u and B=2
                = (u + 2)(u - 2)    Using the difference of squares formula

ANSWER: (u + 2)(u - 2)


QUESTION 3: Factoring a quadratic with leading coefficient greater than 1(Problem type 1)

                Factor: 3y^2 – 4y -7

EXPLANATION: In this question we will factor using a method often called Trial and Error
                The coefficient of y^2 is 3
                Using whole numbers, only 1 and 3 have a product of 3
                So we'll look for integers n and m that satisfy the following
                3y^2 – 4y -7 = (1y + m)(3y + n)
                Using FOIL to expand the right-hand side of this equation, we get the following
                m*n = -7 and n + 3m = -4
                We'll find all integers m and n such that n*m = -7
                Then we'll check to see if n + 3m = -4

n             m            n+3m
1              -7            -4
-1            7              4
-7            1              -20
7              -1            20

From the table, we find that m = 1 and n = -7 work
So we have the following:
3y^2 – 4y -7 = (y + m)(3y + n)
= (y + 1)(3y – 7)

ANSWER: (y + 1)(3y – 7)



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