# 9.1: Addition and Subtraction of Polynomials

## Learning Objectives

At the end of this lesson, students will be able to:

- Write a polynomial expression in standard form.
- Classify polynomial expression by degree.
- Add and subtract polynomials.
- Problem-solve using addition and subtraction of polynomials.

## Vocabulary

Terms introduced in this lesson:

- polynomial
- term
- coefficient
- constant
- degree
- cubic term, quadratic term, linear term
- nth order term
- monomial, binomial
- standard form
- leading term
- leading coefficient
- rearranging terms
- like terms, collecting like terms

## Teaching Strategies and Tips

There are a large number of new terms in this lesson. Introduce new vocabulary with concrete, specific examples. It is also helpful to provide examples of what the new word does *not* mean.

- Polynomials consist of terms with variables of nonnegative integer powers. Polynomials can have more than one variable.

Examples:

These are polynomials:

These are not polynomials:

Have students explain their answers. Suggest that they use explanations such as:

This is not a polynomial because...

- ...it has a negative exponent.
- ...it has a radical.
- ...the power of appears in the denominator.
- ...it has a fractional exponent.
- ...it has an exponential term.

- Terms are added or subtracted “pieces” of the polynomial.

Examples:

The polynomial has terms.

The polynomial has terms; it is called a *tri*nomial.

is a *bi*nomial because it has terms.

and are term poynomials and are called *mono*mials.

In the polynomial , the is a term; but neither nor are terms. is another term; but is not. is a term.

- The constant term is that number appearing by itself without a variable.

Examples:

In the polynomial , the is the only constant term.

The polynomial has no constant terms.

- Coefficients are numbers appearing in terms in front of the variable.

Examples:

. The coefficient of the first term is . The coefficient of the second term is .

. The coefficient of each of the terms is .

- In standard form, a polynomial is arranged in decreasing order of powers; terms with higher exponents appear to the left of other terms.

Examples:

These polynomials are in standard form:

These polynomials are not in standard form:

- The first term of a polynomial in standard form is called the leading term, and the coefficient of the leading term is called the leading coefficient.

Examples:

The leading term and leading coefficient of the polynomial are and , respectively.

The leading term and leading coefficient of the polynomial are and , respectively. Remind students to write polynomials in standard form.

- Like terms are terms with the same variable(s) to the same exponents. Like terms may have different coefficients. A polynomial is simplified if it has no terms that are alike.

Examples:

These are like terms:

and

and

and

and

These are not like terms:

and

and

The polynomial is not simplified.

- The degree of a term is the power (or the sum of powers) of the variable(s). The constant term has a degree of . The degree of a polynomial is the degree of its leading term. Encourage students to name polynomials by their degrees: cubic, quadratic, linear, constant.

Examples:

The term has degree .

The term has degree .

is a fifth-degree polynomial.

is a cubic polynomial. Remind students to write polynomials in standard form.

Assess student vocabulary by asking them to determine all parts (terms, leading term, coefficients, leading coefficient, constant term) of a given polynomial and have them describe it in as many ways as they can (its degree, whether it is in standard form, number of variables, etc.) See Example 1-3.

When adding or subtracting polynomials, suggest that students do so vertically. The vertical or column format helps students keep terms organized.

Example:

*Subtract and simplify.*

Solution: Subtract vertically. Keep like terms aligned.

When simplifying like terms, suggest that students rearrange the terms into *groups* of like terms first. This is especially helpful in *Review Questions* 11, 12, and 16. See also Example 4.

## Error Troubleshooting

General Tip: Remind students to distribute the minus sign to *every* term in the second polynomial when subtracting two polynomials. See Example 6 and *Review Questions* 13-16.

When simplifying polynomials, such as in Example 4b and *Review Questions* 12 and 16, remind students that like terms must have the same variables and exponents.

In Example 6, remind students that to *subtract* *from* means and not the other way around.

Example:

*Subtract* from .

Hint: Setup the problem as . Then distribute the negative inside the parentheses to every term. Group like terms.

General Tip: Some students will give the incorrect degree of a polynomial; remind students write polynomials in standard form and then look for the leading term.

General Tip: Students can check their answers by plugging in a simple value for the variable in the original polynomials and simplified polynomial and check if the results have the same value.

Example:

*Subtract* from .

Solution:

Distribute.

Group like terms. .

Answer:

Check.

Let and .

Original:

Simplified: