Adding and Subtracting Polynomials - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 8. Adding and Subtracting Polynomials

In this chapter, you learn how to add and subtract polynomials. It begins with a discussion of the elementary concepts that you need to know to ensure your success when working with polynomials.

Terms and Monomials

In an algebraic expression, terms are the parts of the expression that are connected to the other parts by plus or minus symbols. If the algebraic expression has no plus or minus symbols, then the algebraic expression itself is a term.

Problem Identify the terms in the given expression.

b. 3x⁵

Solution

Step 1. The expression contains plus and minus symbols, so identify the quantities between the plus and minus symbols.

The terms are –8xy³, , and 27.

b. 3x⁵

Step 1. There are no plus or minus symbols, so the expression is a term.

The term is 3x⁵.

In monomials, no variable divisors, negative exponents, or fractional exponents are allowed.

A monomial is a special type of term that when simplified is a constant or a product of one or more variables raised to nonnegative integer powers, with or without an explicit coefficient.

Problem Specify whether the term is a monomial. Explain your answer.

a. –8xy³

c. 0

d. 3x⁵

e. 27

f. 4x^–3y²

Solution

a. –8xy³

Step 1. Check whether –8xy³ meets the criteria for a monomial.

–8xy³ is a term that is a product of variables raised to positive integer powers, with an explicit coefficient of –8, so it is a monomial.

Step 1. Check whether meets the criteria for a monomial.

is a term, but it contains division by a variable, so it is not a monomial.

c. 0

Step 1. Check whether 0 meets the criteria for a monomial.

0 is a constant, so it is a monomial.

d. 3x⁵

Step 1. Check whether 3x⁵ meets the criteria for a monomial.

3x⁵ is a term that is a product of one variable raised to a positive integer power, with an explicit coefficient of 3, so it is a monomial.

e. 27

Step 1. Check whether 27 meets the criteria for a monomial.

27 is a constant, so it is a monomial.

Step 1. Check whether 4x^{– 3} y² meets the criteria for a monomial.

4x^{– 3} y² contains a negative exponent, so it is not a monomial.

Step 1. Check whether meets the criteria for a monomial.

contains a fractional exponent, so it is not a monomial.

Polynomials

A polynomial is a single monomial or a sum of monomials. A polynomial that has exactly one term is a monomial. A polynomial that has exactly two terms is a binomial. A polynomial that has exactly three terms is a trinomial. A polynomial that has more than three terms is just a general polynomial.

Problem State the most specific name for the given polynomial.

Solution

Step 1. Count the terms of the polynomial.

has exactly two terms.

Step 2. State the specific name.

is a binomial.

Step 1. Count the terms of the polynomial.

has exactly two terms.

Step 2. State the specific name.

is a binomial.

Step 1. Count the terms of the polynomial.

has exactly three terms.

Step 2. State the specific name.

is a trinomial.

Step 1. Count the terms of the polynomial.

has exactly one term.

Step 2. State the specific name.

is a monomial.

Step 1. Count the terms of the polynomial.

has exactly six terms.

Step 2. State the specific name.

is a polynomial.

Like Terms

Monomials that are constants or that have exactly the same variable factors (i.e., the same letters with the same corresponding exponents) are like terms. Like terms are the same except, perhaps, for their coefficients.

Problem State whether the given monomials are like terms. Explain your answer.

Solution

Step 1. Check whether meet the criteria for like terms.

are like terms because they are exactly the same except for their numerical coefficients.

Step 1. Check whether meet the criteria for like terms.

are not like terms because the corresponding exponents on x and y are not the same.

c. 100 and 45

Step 1. Check whether 100 and 45 meet the criteria for like terms.

100 and 45 are like terms because they are both constants.

d. 25 and 25x

Step 1. Check whether 25 and 25x meet the criteria for like terms.

25 and 25x are not like terms because they do not contain the same variable factors.

Finally, monomials that are not like terms are unlike terms.

Addition and Subtraction of Monomials

Because variables are standing in for real numbers, you can use the properties of real numbers to perform operations with polynomials.

Addition and Subtraction of Monomials

1. To add monomials that are like terms, add their numerical coefficients and use the sum as the coefficient of their common variable component.

2. To subtract monomials that are like terms, subtract their numerical coefficients and use the difference as the coefficient of their common variable component.

3. To add or subtract unlike terms, indicate the addition or subtraction.

Problem Simplify.

Solution

Step 1. Check for like terms.

Step 2. Add the numerical coefficients.

Step 3. Use the sum as the coefficient of x.

. In addition and subtraction, the exponentson the variables do not change.

Step 1. Check for like terms.

4x²y³and 7x³y² are not like terms, so leave the problem as indicated subtraction: .

Step 1. Check for like terms.

Step 2. Combine the numerical coefficients.

Step 3. Use the result as the coefficient of x².

Step 1. Check for like terms.

25 + 25x

25 and 25x are not like terms, so leave the problem as indicated addition: 25 + 25x.

These are not like terms, so you cannot combine them into one single term.

Step 1. Check for like terms.

Step 2. Subtract the numerical coefficients.

Step 3. Use the result as the coefficient of x².

Combining Like Terms

When you have an assortment of like terms in the same expression, systematically combine matching like terms in the expression. (For example, you might proceed from left to right.) To organize the process, use the properties of real numbers to rearrange the expression so that matching like terms are together (later, you might choose do this step mentally). If the expression includes unlike terms, just indicate the sums or differences of such terms. To avoid sign errors as you work, keep a – symbol with the number that follows it.

Problem Simplify

Solution

Step 1. Check for like terms.

The like terms are 4x³, and 2x³,5x²and 7x², and 25 and 5.

Step 2. Rearrange the expression so that like terms are together.

Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

When you are simplifying, rearrange so that like terms are together can be done mentally. However, writing out this step helps you avoid careless errors.

Because + – is equivalent to –, it is customary to change + – to simply – when you are simplifying expressions.

Step 4. Review the main results.

Addition and Subtraction of Polynomials

Addition of Polynomials

To add two or more polynomials, add like monomial terms and simply indicate addition or subtraction of unlike terms.

Problem Perform the indicated addition.

Solution

Step 1. Remove parentheses.

Step 2. Rearrange the terms so that like terms are together. (You might do this step mentally.)

Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

You should write polynomial answers in descending powers of a variable.

Step 4. Review the main results.

Step 1. Remove parentheses.

Step 2. Rearrange the terms so that like terms are together. (You might do this step mentally.)

Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

Step 4. Review the main results.

Subtraction of Polynomials

To subtract two polynomials, add the opposite of the second polynomial.

You can accomplish subtraction of polynomials by enclosing both polynomials in parentheses and then placing a minus symbol between them. Of course, make sure that the minus symbol precedes the polynomial that is being subtracted.

Problem Perform the indicated subtraction.