Polynomials - Easy Mathematics Step-by-Step

Easy Mathematics Step-by-Step (2012)

Chapter 13. Polynomials

In this chapter, you learn about polynomials. This chapter 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.

Solution

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

The terms are , and 27.

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

The term is 3x⁵.

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.

In monomials, no variable divisors, negative exponents, or variables as radicands of simplified radicals are allowed.

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

c. 0

d. 3x⁵

f. 4x^–3y²

Solution

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

–8x is a term that is a variable raised to a positive integer power of 1 (understood), 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.

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

3x⁵ is a term that is a variable raised to a positive integer power of 5, with an explicit coefficient of 3, so it is a monomial.

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

is a constant, so it is a monomial.

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

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

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

is a term, but it contains a variable as the radicand of a simplified radical, so it is not a monomial.

The constants in monomials can be divisors, have negative exponents, or be radicands in a radical. For instance, is 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 monomials 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. Terms that are not like terms are unlike terms.

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

c. 100 and 45

Solution

a.
and

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

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

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

and 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.

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.

Adding and Subtracting Monomials

Because variables are standing in for numbers, you rely on the properties of numbers to justify operations with polynomials. (See Chapter 1 for a discussion of the properties of numbers.)

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.

and are like terms.

Step 2. Add the numerical coefficients.

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

In addition and subtraction, the exponents on your variables do not change.

Step 1. Check for like terms.

and are not like terms, so leave the problem as indicated subtraction:

Step 1. Check for like terms.

, , and are like terms.

Step 2. Combine the numerical coefficients.

Step 3. Use the result as the coefficient of

Step 1. Check for like terms.

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

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

Step 1. Check for like terms.

and are like terms.

Step 2. Subtract the numerical coefficients.

Step 3. Use the result as the coefficient of

Simplifying Polynomial Expressions

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.) You are simplifying the expression when you do this. To organize the process, use the properties of 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 and , and , and 25 and 5.

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

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

Remember, when rearranging, to keep a – symbol with the number that follows it.

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

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

Step 4. Review the main results.

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

Step 1. Check for like terms.

The like terms are , , , and and 30, 100, and 25.

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.

Step 4. Review the main results.

Adding Polynomials

Addition of polynomials involves adding like terms.

Addition of Polynomials

To add two or more polynomials, add like monomial terms and simply indicate addition 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 may 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.

Step 1. Remove parentheses.

Step 2. Rearrange the terms so that like terms are together. (You may 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.

Subtracting Polynomials

Subtraction of polynomials relies on your skills in adding polynomials.

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.