Home
Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Inequalities
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Polynomials
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Conjugates
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Formulas
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Roots
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
 
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Factoring Trinomials by Grouping

Factoring a Trinomial of the Form ax2 + bx + c by Grouping

In this method, we rewrite the middle term of the trinomial using two terms. Then we factor by grouping.

For example, we’ll use this method to factor 15x2 - 16x + 4.

Write the middle term, -16x, as -10x - 6x.

Group the first two terms and group the last two terms.

Factor 5x out of the first group; factor -2 out the second group.

Finally, factor out (3x - 2).

15x2 - 16x + 4

= 15x2 - 10x - 6x + 4

= (15x2 - 10x) + (-6x + 4)

= 5x(3x - 2) + (-2)(3x - 2)

= (3x - 2)(5x - 2)

 

The key to this method is knowing how to rewrite the middle term of the trinomial. The following procedure describes a way to do this.

 

Procedure — To Factor ax2 + bx + c by Grouping

Step 1 Factor out common factors (other than 1 or -1).

Step 2 List the values of a, b, and c. Then find two integers whose product is ac and whose sum is b. If no two such integers exist then the trinomial is not factorable over the integers.

Step 3 Replace the middle term, bx, with a sum or difference using the two integers found in Step 2.

Step 4 Factor by grouping. To check the factorization, multiply the binomial factors.

 

Note:

This method also works when a = 1. However, in those cases The Product-Sum method requires fewer steps.

 

Example 1

Factor: 6x2 + 7x + 2.

Solution

Step 1 Factor out common factors (other than 1 or -1).

 There are no common factors other than 1 and -1.

Step 2 List the values of a, b, and c. Then find two integers whose product is ac and whose sum is b.

6x2 + 7x + 2 has the form ax2 + bx + c where a = 6, b = 7, and c = 2.

The product ac is 6 · 2 = 12.

Thus, find two integers whose product, ac, is 12 and whose sum, b, is 7.

• Since their product is positive, the integers must have the same sign.

• Since their sum is also positive, the integers must both be positive.

Here are the possibilities:

Product

1 · 12

2 · 6

3 · 4

Sum

13

8

7

The integers 3 and 4 satisfy the requirements that their product is 12 and their sum is 7.
Step 3 Replace the middle term, bx, with a sum or difference using the two integers found in Step 2. 6x2 + 7x + 2
Replace 7x with 3x + 4x. = 6x2 + 3x + 4x + 2
Step 4 Factor by grouping.

Group the first pair of terms and group the second pair of terms.

Factor 3x out of the first group; factor 2 out of the second group.

Factor out the common factor (2x + 1).

= (6x2 + 3x) + (4x + 2)

= 3x(2x + 1) + 2(x + 1)

= (2x + 1)(3x + 2)

 

The result is: 6x2 + 7x + 2 = (2x + 1)(3x + 2). You can multiply to check the factorization. We leave the check to you.

Note:

We replaced 7x with 3x + 4x.

If we switch 3x and 4x, we can still group and factor:

= 6x2 + 4x + 3x + 2

= (6x2 + 4x) + (3x + 2)

= 2x(3x + 2) + 1(3x + 2)

= (3x + 2)(2x + 1)

 
All Right Reserved. Copyright 2005-2024