The document provides examples of factoring polynomials by finding the greatest common factor (GCF). It explains that the GCF is the product of all prime factors that are common to each term, with each factor raised to the highest power that it occurs. Then, it works through 5 examples of factoring polynomials by identifying the GCF and factoring it out of each expression.

2. The Greatest Monomial Factor of
two or more terms is the product
of all prime algebraic factors to
every term, each to the highest
power that it occurs in all the terms.
Check your answer by multiplying
the factors.

3. Example 1. Factor out 4x2 + 6x3.
Find first the GCF (Greatest Common Factor). Observe the following.
4x2 = 2 . 2 . x. x
6x3 = 2 . 3 . x. x . x
= 2x2
The GCF is 2x2 .
Factor out 2x2 from the given polynomial. = 4x2 + 6x3
= 2x2( 2 + 3x)
Hence, the factored form of 4x2 + 6x3 is 2x2(2 + 3x).

4. Example 2. Factor out a2 + 2a – a2b.
Find the GCF. a2 = a. a
2a = 2 . a
a2b = a . a . b
= a
The GCF is a.
Factor out a from the given polynomial. = a2 + 2a – a2b
= a(a + 2 – ab)
Hence the factored form a2 + 2a – a2b is a(a + 2 – ab).

5. Example 3. Factor out 5x3 + 3x5.
Find the GCF. 5x3 = 5 . x. x. x
3x5 = 3 . x . x. x. x. x
= x3
The GCF is x3.
Factor out x3 from the given polynomial. = 5x3 + 3x5
= x3(5 + 3x2)
Hence the factored form of 5x3 + 3x5 is x3(5 + 3x2).

6. Example 4. Factor 15a2b – 10ab.
Find the GCF. 15a2b = 5. 3 . a. a. b
10ab = 5. 2 . a . b
= 5ab
The GCF is 5ab.
Factor out the GCF from the given polynomial. = 15a2b – 10ab
= 5ab(3a – 2)
Hence the factored form is 5ab(3a – 2).

7. Example 5. Factor 16x2y + 10xy2 – 8xy
Find the GCF. 16x2y = 2 . 2 . 2 . 2 . x . x . y
10xy2 = 5 . 2 . x. y . y
8xy = 2 . 2 . 2 . x . Y
= 2xy
The GCF is 2xy.
Factored out the GCF from the given factor. = 16x2y + 10xy2 – 8xy
= 2xy(8x + 5y – 4)
Hence the factored form is 2xy(8x + 5y – 4).