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# 9.1: Addition and Subtraction of Polynomials

Difficulty Level: 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.
• 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
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.
...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 trinomial.

\begin{align*}x^2 + y^2\end{align*} is a binomial 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 monomials.

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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