# 9.1: Addition and Subtraction of Polynomials

**At Grade**Created by: CK-12

## 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:

\begin{align*}& -\sqrt{12}x^8 - x^5 + \pi\\ & x^5 + x^4 - x^3 + x^2 - x + 1\\ & -\frac{3} {7}\\ & x^2 + y^2\\ & xy\\ & 2x^2 - 4xy + 1\end{align*}

These are not polynomials:

\begin{align*}& \sqrt{x} + x + 2\\ & y^{4.2} - x^{3.5} + x^{1.1} - x + 1\\ & x + \frac{1} {x} - 3\\ & 2^x + 8\\ & x^{-2} + x^{-1} + 1\end{align*}

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 \begin{align*}x\end{align*} 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 \begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*} has \begin{align*}6\end{align*} terms.

The polynomial \begin{align*}2x^2 - 4xy + 1\end{align*} has \begin{align*}3\end{align*} terms; it is called a *tri*nomial.

\begin{align*}x^2 + y^2\end{align*} is a *bi*nomial because it has \begin{align*}2\end{align*} terms.

\begin{align*}xy\end{align*} and \begin{align*}-\frac{3} {7}\end{align*} are \begin{align*}1-\end{align*}term poynomials and are called *mono*mials.

In the polynomial \begin{align*}-2x^5 + 7x^3 - x + 8\end{align*}, the \begin{align*}8\end{align*} is a term; but neither \begin{align*}-2, 5, 7,\end{align*} nor \begin{align*}3\end{align*} are terms. \begin{align*}-x\end{align*} is another term; but \begin{align*}x^3\end{align*} is not. \begin{align*}7x^3\end{align*} is a term.

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

Examples:

In the polynomial \begin{align*}-2x^5 + 7x^3 - x + 8\end{align*}, the \begin{align*}8\end{align*} is the only constant term.

The polynomial \begin{align*}x^2 + y^2\end{align*} has no constant terms.

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

Examples:

\begin{align*}2x^2 - 4xy + 1\end{align*}. The coefficient of the first term is \begin{align*}2\end{align*}. The coefficient of the second term is \begin{align*}-4\end{align*}.

\begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*}. The coefficient of each of the terms is \begin{align*}1\end{align*}.

- 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:

\begin{align*} & x^5 + x^4 - x^3 + x^2 - x + 1\\ & xy + x - y - 1\end{align*}

These polynomials are not in standard form:

\begin{align*} & 1 - x + x^2 - x^3 + x^4 + x^5\\ & xy + x^2 y^2 - 1\end{align*}

- 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 \begin{align*}2x^2 - 4xy + 1\end{align*} are \begin{align*}2x^2\end{align*} and \begin{align*}2\end{align*}, respectively.

The leading term and leading coefficient of the polynomial \begin{align*}9x^2 + 8x^3 + x + 1\end{align*} are \begin{align*}8x^3\end{align*} and \begin{align*}8\end{align*}, 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:

\begin{align*}2x^3\end{align*} and \begin{align*}-8x^3\end{align*}

\begin{align*}-xy\end{align*} and \begin{align*}17.2xy\end{align*}

\begin{align*}2x, -4x,\end{align*} and \begin{align*}\sqrt{2}x\end{align*}

\begin{align*}-4, \pi,\end{align*} and \begin{align*}\sqrt{2}\end{align*}

These are not like terms:

\begin{align*}x^2y\end{align*} and \begin{align*}xy^2\end{align*}

\begin{align*}x^2y^2\end{align*} and \begin{align*}x^2 + y^2\end{align*}

The polynomial \begin{align*}-x^3 + 3.1x^2 - 4x^2 + x - 2\end{align*} 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 \begin{align*}0\end{align*}. 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 \begin{align*}-8x^3\end{align*} has degree \begin{align*}3\end{align*}.

The term \begin{align*}7.1x^2y^2\end{align*} has degree \begin{align*}4\end{align*}.

\begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*} is a fifth-degree polynomial.

\begin{align*}9x^2 + 8x^3 + x + 1\end{align*} 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.*

\begin{align*}4x^2 + 2x + 1 - (3x^2 + x - 4)\end{align*}

Solution: Subtract vertically. Keep like terms aligned.

\begin{align*}& \quad 4x^2 + 2x + 1\\ & -3x^2 - x + 4\\ & \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ & \qquad x^2 + x + 5\end{align*}

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* \begin{align*}A\end{align*} *from* \begin{align*}B\end{align*} means \begin{align*}B - A\end{align*} and not the other way around.

Example:

*Subtract* \begin{align*}-2m^2 + 3n^2 + 4mn - 1\end{align*} from \begin{align*}-2n^2 - 7 + 2mn + 8m^2\end{align*}.

Hint: Setup the problem as \begin{align*}-2n^2 - 7 + 2mn + 8m^2 -(-2m^2 + 3n^2 + 4mn - 1)\end{align*}. 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* \begin{align*}-2m^2 + 3n^2 + 4mn - 1\end{align*} from \begin{align*}-2n^2 - 7 + 2mn + 8m^2\end{align*}.

Solution:

Distribute. \begin{align*}-2n^2 - 7 + 2mn + 8m^2 - (-2m^2 + 3n^2 + 4mn - 1)\end{align*}

Group like terms. \begin{align*}-2n^2 - 7 + 2mn + 8m^2 + 2m^2 - 3n^2 - 4mn + 1\end{align*}.

\begin{align*}(8m^2 + 2m^2) + (-2n^2 - 3n^2) + (2mn - 4mn) + (-7 + 1)\end{align*}

Answer: \begin{align*}10m^2 - 5n^2 - 2mn - 6\end{align*}

Check.

Let \begin{align*}m = -1\end{align*} and \begin{align*}n = 1\end{align*}.

Original: \begin{align*}-2n^2 - 7 + 2mn + 8m^2 - (-2m^2 + 3n^2 + 4mn - 1)\end{align*}

\begin{align*}& -2 \cdot 1^2 - 7 + 2 \cdot (-1) \cdot 1 + 8 \cdot (-1)^2 - (-2 \cdot (-1)^2 + 3 \cdot 1^2 + 4 \cdot (-1) \cdot 1 - 1)\\ & -3 - (-4) = 1\end{align*}

Simplified: \begin{align*}10m^2 - 5n^2 - 2mn - 6\end{align*}

\begin{align*}& 10 \cdot (-1)^2 - 5 \cdot 1^2 - 2 \cdot (-1) \cdot 1 - 6\\ & 10 - 5 + 2 - 6 = 1\end{align*}

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