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June 19th

June 19th

Greatest Common Factor and Factoring by Grouping

(Review) Factoring
Definition:
A factor is a number, variable, monomial, or polynomial which is multiplies by
another number, variable, monomial, or polynomial to obtain a product.

1. List all the possible factors of the fol lowing numbers :
a. 12
b. 32
c. 19
d. -45
In the above, the number 19 is an example of a ____________ number because its only positive
factors are one and itself .

(Review) Greatest Common Factor
Definition: The greatest common factor of two or more numbers is the largest number that
divides (goes into) the given numbers with a remainder of zero .

2. Find the GCF (greatest common factor) of the following numbers:
a. 36 and 54
b. 15 and 60
c. 21, 42, and 63
d. 28 and 39

Greatest Common Factor of Polynomials
In order to find the GCF of two or more monomials ,
I. Find the GCF of the coefficients;
II. Find the GCF of the variables;
III. Rewrite the GCF as a product of the GCF of the coefficients times the GCF of the variables.

Examples:
1. Find the GCF of x4 and x7.

Step I : The only coefficients are 1’s, so this is the GCF of the coefficients.

Step II: Rewrite the two monomials as products of xs without using exponents :
x4 = x · x · x · x
x7 = x · x · x · x · x · x · x
Since each monomial has 4 xs in it, the GCF of the variables is x 4.
Step III: The GCF of x4 and x7 is x4.

2. Find the GCF of xy2 and x5y4.
Step I: The only coefficients are 1’s, so this is the GCF of the coefficients.

Step II: Rewrite the two monomials as products of xs and ys without using exponents:
xy2= x · y · y
x5y4 = x · x · x · x · x · y · y · y · y
Since each monomial has 1 x and 2 ys in it, the GCF of the variables is xy2.

Step III: The GCF of xy2 and x5y4 is xy2.

3. Find the GCF of 24x4y and 9x7y4.

Step I: The GCF of the coefficients 24 and 9 is 3.

Step II: Rewrite the two monomials as products of xs and ys without using exponents:
x4y = x · x · x · x · y
x7y4 = x · x · x · x · x · x · x · y · y · y · y
Since each monomial has 4 xs and 1 y in it, the GCF of the variables is x4y.

Step III: The GCF of 24x4y and 9x7y4 is 3x4y.

3. Find the GCF of 14xy and 21x5y2.

4. Find the GCF of 18c9 and -6c4x.

5. Find the GCF of 23a6b3 and 42a3c2.

Factoring
To factor a polynomial, an attempt should be made to find the GCF of the monomials in the
polynomial. Then this GCF should be factored out of the polynomial by “undistributing” the
GCF out of all the monomials in the polynomial. Note that if the leading coefficient is negative ,
then the GCF should also be negative.

Examples:
1. Factor 7x + 14y.
Step I: Find the GCF of the coefficients. GCF(7, 14) = 7.

Step II: Find the GCF of the variable parts. There is none, so nothing changes .

Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:

2. Factor 8x3 + 6x2.
Step I: Find the GCF of the coefficients. GCF(8, 6) = 2.

Step II: Find the GCF of the variable parts. GCF(x3, x2) = x2.

Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:

3. Factor 18y3(x – 1) + 21y(x – 1)
Step I: Find the GCF of the coefficients: GCF(18, 21) = 3.
Step II: Find the GCF of the variable parts: GCF(y3(x – 1), y(x – 1)) = y(x – 1)
Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:

6. Factor 3ab + 7b.

7. Factor -28a3b7 – 36a2b5.

8. Factor 8a2bc – 12ab2c.

9. Factor 9xy3 + 27xy – x5. (Hint: What is the leading coefficient?)

10. Factor (9 – 3x)(27y2) – (9 – 3x)(15y3)

Factoring by Grouping
If a polynomial contains four or more terms, it may be helpful to put the terms into groups of two
and factor out a common factor from each of these groups. This is called grouping.

Examples:
1. Factor x2y + 6x + 3xy2 + 18y

Step I: Group the terms so that each group shares a common factor:
x2y + 6x + 3xy2 + 18y = (x2y + 6x) + (3xy2 + 18y)

Step II: Factor out the common terms from each group:
(x2y + 6x) = x(xy + 6)
(3xy2 + 18y) = 3y(xy + 6)

Step III: Rewrite the polynomial as the sum of the factored groups:
x2y + 6x + 3xy2 + 18y = x(xy + 6) + 3y(xy + 6)
Step IV: Factor the resulting polynomial from Step III:
x(xy + 6) + 3y(xy + 6) = (xy + 6)(x + 3y)

2. Factor 2x3 + 3x2 + 2x + 3

Step I: Group the terms so that each group shares a common factor:
2x3 + 3x2 + 2x + 3 = (2x3 + 3x2) + (2x + 3)

Step II: Factor out the common terms from each group:
(2x3 + 3x2) = x2(2x + 3)
(2x + 3) = 1(2x + 3) (Note: There’s no common term, so the GCD is 1.)

Step III: Rewrite the polynomial as the sum of the factored groups from Step II.
2x3 + 3x2 + 2x + 3 = x2(2x + 3) + 1(2x + 3)

Step IV: Factor out the resulting polynomial from Step III:
x2(2x + 3) + 1(2x + 3) = (x2 + 1)(2x + 3)

3. Factor 3x3 + 3x2 – 4x – 4.

Step I: Group the terms so that each group shares a common factor:€
3x3+ 3x2 – 4x – 4 = (3x3 + 3x2) + (-4x – 4)

Step II: Factor out the common terms from each group:
(3x3 + 3x2) = 3x2(x + 1)
(-4x – 4) = -4(x + 1)

Step III: Rewrite the polynomial as the sum of the factored groups from Step II.
3x3 + 3x2 – 4x – 4 = 3x2(x + 1) + (-4(x + 1)) = 3x2(x + 1) – 4(x + 1)

Step IV: Factor out the resulting polynomial from Step III:
3x2(x + 1) – 4(x + 1) = (3x2 – 4)(x + 1)

11. Factor 4a3 + 8a + 3a2b2 + 6b2.

12. Factor 3a2x + a2y – 12x – 4y.

13. 6ax + 6xb + 10ay + 10yb.

14. 6x + 9xy + 10y + 15y2.

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