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Factoring by Grouping

Factoring:

Some rules governing factoring in elementary algebra:

1. The numbers (coefficients) must be real and rational.

2. The powers must be whole numbers.

3. Prime factors include:

a) Linear polynomial, such as (x + 3), (x + y + z)

b) The sum of an even degree term (2 or higher) with any real number, such as:

(x2 + 4), (x2 y2  + 9), (2x2  + 25), (x2  + 3), etc.

c) The sum of two even-powered terms, such as: (x4 + y4), (x6 + y8), (x2 + y6)

d) Certain quadratic trinomials which cannot be expressed as the product of two linear binomials with rational numbers. There is a special test for this which will be discussed later.

Factor Quadratic Polynomials:

WHAT TO DO: HOW TO DO IT:
1. First, examine the polynomial to see if it has a common factor(s) in each term. Integers as common factors are left in composite form and letters are left in power form. Note the remaining factor is prime. a) 3x2 + 12

    3(x2 + 4) prime factors

b) 5x3 + 7x2

     x2(5x + 7) prime factors

Factor out that common factor(s) and see what is left. If the remaining polynomial is prime*, leave the factors “as is”. c) 3x3 − 21x2 + 27x

3x(x2 − 7x + 9) prime factors

If the inner polynomial is factorable, set it aside to examine separately and come back to complete. d) 4x2 − 12x − 8x + 24

    4(x2 − 3x − 2x + 6)

2. Consider the inner polynomial: x2 − 3x − 2x + 6 grouped 2×2 by underlining the first two and last two. Factor common factor from each group.

Bring down middle sign.

Complete factoring

 

Now, go back to the original problem and write the answer in completely factored form. d) 4x2 − 20x + 24

    4(x2 − 5x + 6)

    4(x − 3)(x − 2)

 

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