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Raising an Exponential Expression to a Power
Horizontal Line Test
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Solving Quadratic Equations by Completing the Square
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Adding and Subtracting Polynomials
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Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
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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
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Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
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
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
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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
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
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
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
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Adding and Subtracting Polynomials

Objective Learn how to add and subtract polynomials.

This lesson is not difficult conceptually, but it is important that you get sufficient practice in addition and subtraction of polynomials.


Adding Polynomials

Let's begin with an example in order to see how to add polynomials.

Example 1

Add x 3 + x + 1 and 3x 3 + x 2 + 2x .


First, write the polynomials side by side.

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

The monomials x 3 and x each occur twice, so group the terms together and combine them.

x 3 + x + 1 + 3x 3 + x 2 + 2x = ( x 3 + 3x 3 ) + x 2 + ( x + 2x ) + 1
  = 4x 3 + x 2 + 3x + 1

This polynomial is simpler than the original two polynomials written side by side.


Like Terms

To be explicit about the steps taken to add polynomials, let's talk a little bit about like terms.

Definition of Like Terms

When we are given two polynomials, we say the monomials in the polynomials are like terms if they contain exactly the same number of occurrences of each variable.

Example 2

Name the like terms in x 4 + 4x 3 + 6x 2 + 4 x + 1 and x 3 + x 2 + 1.


There are several pairs of like terms, shown in the diagram below.

The pairs of like terms are connected by arrows. So, the like terms are 4x 3 and x 3 , 6x 2 and x 2 , 1 and 1.


Example 3

Name the like terms in ab + 2a + 2b and ab - a + 1.


In this case, there are two pairs of like terms, ab and ab , 2a and -a , shown in the diagram below.

Whenever there are like terms, collect them and add them together to get a single term. In Example 2, add 4x 3 and x 3 together to get 5x 3 . In the same way, add 6x 2 and x 2 together to get 7x 2 , and finally, 1 + 1 = 2.

When we do this, we get a simpler polynomial.

( x 4 + 4x 3 + 6x 2 + 4 x + 1 ) + ( x 3 + x 2 + 1 ) = x 4 + 5x 3 + 7x 2 + 4x + 2

In Example 3, collect the ab and a terms to get

( ab + 2a + 2b ) + ( ab - a + 1) = 2ab + a + 2b + 1.


Key Idea

Sums of polynomials can be simplified by adding together like terms. In the same way, differences of polynomials can be simplified by collecting together all pairs of like terms.


Subtracting Polynomials

Example 4

Simplify ( x 3 - 3x 2 + 3 x - 1) - ( x 2 + 2x + 1).


First write this as an addition expression by adding the additive inverse.

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

To simplify the difference, collect all pairs of like terms. These are shown in the diagram below.

Collect the terms to get the following polynomial.

x 3 + ( - 3 - 1) x 2 + (3 - 2) x + ( - 1 - 1) = x 3 - 4x 2 + x - 2

There is another method of adding and subtracting polynomials that is similar to adding and subtracting numbers. Write the polynomials one over the other, with like terms lined up in columns. To find the difference, either subtract the coefficients in each of the columns, or add the additive inverse.


Example 5

Subtract 4x 3 + 5x 2 + 6x + 7 and 2x 3 + 3x + 4.


Write the polynomials with the like terms in columns.

Notice that a zero coefficient is added for each missing term. Now add the coefficients in each column.

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