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GCF & Factor by Grouping. MATH 018 Combined Algebra S. Rook. Overview. Section 6.1 in the textbook: Finding the Greatest Common Factor (GCF) of a list of numbers Finding the GCF of a list of terms Factoring out the GCF from a polynomial Factoring by grouping.

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Gcf factor by grouping

GCF & Factor by Grouping

MATH 018

Combined Algebra

S. Rook


Overview
Overview

  • Section 6.1 in the textbook:

    • Finding the Greatest Common Factor (GCF) of a list of numbers

    • Finding the GCF of a list of terms

    • Factoring out the GCF from a polynomial

    • Factoring by grouping



Greatest common factor gcf
Greatest Common Factor (GCF) Numbers

Greatest Common Factor (GCF): the largest product of coefficients and variables that is common to (evenly divisible by) all terms of a polynomial

Since the GCF must divide evenly into ALL the numbers, it can only be as large as the SMALLEST number in the polynomial

4


Finding the gcf of a list of numbers
Finding the GCF of a List of Numbers Numbers

  • To find the GCF of a list of numbers (e.g. 16 and 24):

    • Start with the lowest number in the list

    • Check to see if that number divides into ALL numbers in the list evenly; if so, this is the GCF

    • If not, decrease the number by 1 and try again

    • If a number cannot be found before 1 is reached, the list of numbers has a GCF of 1

  • Mastering how to find the GCF of the coefficients (numbers) WILL take practice!

  • Quicker to use prime factorization

5


Finding the gcf of a list of numbers example
Finding the GCF of a List of Numbers (Example) Numbers

Ex 1: Find the GCF of the list of numbers:

a) 4 and 8

b) 7 and 15

c) 12, 30, and 36



Finding the gcf of a list of terms1
Finding the GCF of a List of Terms Numbers

Much easier to find the GCF of variables:

IF each term contains at least one instance of the variable:

Take the variable with the lowest exponent since ALL terms will AT LEAST have this in common

IF a variable is not present in ALL the terms:

That variable CANNOT be a part of the GCF since it is not COMMON to ALL the terms!

8


Finding the gcf of a list of terms example
Finding the GCF of a List of Terms (Example) Numbers

Ex 2: Find the GCF:

a) 4x2, 6x3, and 8x

b) 2r2s4, 5r5s3, 10r9s2

c) 14a5, 21a3b4, 28b3



Factoring out the gcf from a polynomial1
Factoring out the GCF from a Polynomial Numbers

The first step when factoring is to ALWAYS check if it is possible to pull a GCF from the polynomial

Recall that the GCF must divide evenly into ALL terms of the polynomial

If there is a GCF, “pull” it out of the polynomial by reversing the Distributive Property

What operation would we use to reverse the Distributive Property?

This is called factoring out the GCF

11


Factoring out the gcf from a polynomial continued
Factoring out the GCF from a Polynomial (Continued) Numbers

By factoring out a GCF, the terms become smaller and thus easier to work with

Sometimes, a polynomial can only be factored after removing the GCF

12


Factoring out the gcf from a polynomial example
Factoring out the GCF from a Polynomial (Example) Numbers

Ex 3: Factor the GCF from the polynomial:

a) 2x3 – 4x2 + 6x

b) 10as2 + 15as – 20a

c) 9x2 + 16

d) 5x(x + 1) + -1(x + 1)



Factoring in general
Factoring in General Numbers

  • When we factor, we are trying to express sums and differences of terms into a product

    • There are cases where working with a polynomial in product form is easier

      • One such case is solving quadratic equations which we will discuss at the end of the chapter

  • We will discuss many factoring strategies in this chapter

    • We choose a factoring strategy based primarily on the number of terms in the polynomial

    • Concentrate on knowing when to apply each factoring strategy


Factoring by grouping1
Factoring by Grouping Numbers

Before factoring, ALWAYS see if a GCF can be removed from all the terms

Used when there are FOUR terms

e.g. x2 – 3x + x – 3

Group the first two terms together and the last two terms together

Factor the GCF out of each pair of groupings

This step results in the situation depicted in Example 3d

What results inside of the parentheses MUST be the same!

16


Factoring by grouping example
Factoring by Grouping (Example) Numbers

Ex 4: Factor the polynomial:

a) 12x2 – 3x + 8x – 2

b) 20rs2 + 25rs + 16rs + 20r


Negative third term
Negative Third Term Numbers

When factoring by grouping AND the THIRD term is NEGATIVE e.g. x2 – 4x – 7x – 28:

Push the negative of the THIRD term into the second grouping

The sign between the two groupings should ALWAYS be a plus

Factor out the NEGATIVE GCF from the second grouping

What results inside of the parentheses MUST be the same!

18


Negative third term example
Negative Third Term (Example) Numbers

Ex 5: Factor the polynomial:

a) 6x2 + 30x – 4x – 20

b) 18x2 – 24x – 3x + 4


Summary
Summary Numbers

  • After studying these slides, you should know how to do the following:

    • Identify the GCF of a list of numbers

    • Identify the GCF of a group of terms

    • Factor the GCF from a polynomial

    • Factor by grouping

  • Additional Practice

    • See the list of suggested problems for 6.1

  • Next lesson

    • Factoring Easy Trinomials (Section 6.2)


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