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# Operations on Polynomials

### Learning Outcomes

• Identify the degree, leading coefficient, and leading term of a polynomial.
• Multiply polynomials.
The area of the front of the doghouse described in the introduction was $4{x}^{2}+\frac{1}{2}x$ ft2. This is an example of a polynomial which is a sum of or difference of terms each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as $384\pi$, is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product ${a}_{i}{x}^{i}$, such as $384\pi w$, is a term of a polynomial. If a term does not contain a variable, it is called a constant. A polynomial containing only one term, such as $5{x}^{4}$, is called a monomial. A polynomial containing two terms, such as $2x - 9$, is called a binomial. A polynomial containing three terms, such as $-3{x}^{2}+8x - 7$, is called a trinomial. We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

### A General Note: Polynomials

A polynomial is an expression that can be written in the form
${a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$
Each real number ais called a coefficient. The number ${a}_{0}$ that is not multiplied by a variable is called a constant. Each product ${a}_{i}{x}^{i}$ is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.

### How To: Given a polynomial expression, identify the degree and leading coefficient

1. Find the highest power of x to determine the degree.
2. Identify the term containing the highest power of x to find the leading term.
3. Identify the coefficient of the leading term.

### Example: Identifying the Degree and Leading Coefficient of a Polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.
1. $3+2{x}^{2}-4{x}^{3}$
2. $5{t}^{5}-2{t}^{3}+7t$
3. $6p-{p}^{3}-2$

1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, $-4{x}^{3}$. The leading coefficient is the coefficient of that term, $-4$.
2. The highest power of t is $5$, so the degree is $5$. The leading term is the term containing that degree, $5{t}^{5}$. The leading coefficient is the coefficient of that term, $5$.
3. The highest power of p is $3$, so the degree is $3$. The leading term is the term containing that degree, $-{p}^{3}$, The leading coefficient is the coefficient of that term, $-1$.

### Try It

Identify the degree, leading term, and leading coefficient of the polynomial $4{x}^{2}-{x}^{6}+2x - 6$.

Answer: The degree is 6, the leading term is $-{x}^{6}$, and the leading coefficient is $-1$.

[ohm_question]93531[/ohm_question]
In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial. https://youtu.be/3u16B2PN9zk

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, $5{x}^{2}$ and $-2{x}^{2}$ are like terms and can be added to get $3{x}^{2}$, but $3x$ and $3{x}^{2}$ are not like terms and therefore cannot be added.

### How To: Given multiple polynomials, add or subtract them to simplify the expressions

1. Combine like terms.
2. Simplify and write in standard form. Standard form means you start with the leading term, and write the rest of the terms in descending order by degree.

Find the sum.

$\left(12{x}^{2}+9x - 21\right)+\left(4{x}^{3}+8{x}^{2}-5x+20\right)$

Answer: [latex-display]\begin{array}{cc}4{x}^{3}+\left(12{x}^{2}+8{x}^{2}\right)+\left(9x - 5x\right)+\left(-21+20\right) \hfill & \text{Combine like terms}.\hfill \\ 4{x}^{3}+20{x}^{2}+4x - 1\hfill & \text{Simplify}.\hfill \end{array}[/latex-display]

#### Analysis of the Solution

We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be the same. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.

### Try It

Find the sum. [latex-display]\left(2{x}^{3}+5{x}^{2}-x+1\right)+\left(2{x}^{2}-3x - 4\right)[/latex-display]

[ohm_question]93536[/ohm_question]

### Be careful When subtracting polynomials

When subtracting a polynomial from another, be careful to subtract each term in the second from the first. That is, use the distributive property to distribute the minus sign through the second polynomial. $\begin{array}{cc}\left(3x^2-2x+9\right)-\left(x^2-4x+5\right)\text{}\hfill &\text{Distribute the negative in front of the parenthesis} \hfill \\ 3x^2-2x+9 -x^2 -\left(-4x\right) - 5\hfill & \text{Be careful when subtracting a negative}.\hfill \\ 3x^2 - x^2 -2x+4x+9-5\hfill & \text{Rearrange terms in descending order of degree} \hfill \\ 2x^2 +2x +4 \hfill & \text{Combine like terms}. \hfill \end{array}$

### Example: Subtracting Polynomials

Find the difference.

$\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)$

Answer: [latex-display]\begin{array}{cc}7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+\left(6x - 3x\right)+\left(1 - 2\right)\text{ }\hfill & \text{Combine like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\hfill & \text{Simplify}.\hfill \end{array}[/latex-display]

#### Analysis of the Solution

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

### Try It

Find the difference. [latex-display]\left(-7{x}^{3}-7{x}^{2}+6x - 2\right)-\left(4{x}^{3}-6{x}^{2}-x+7\right)[/latex-display]

[ohm_question]93537[/ohm_question]
Watch this video to see more examples of adding and subtracting polynomials. https://youtu.be/jiq3toC7wGM

## Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

## Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial we use the distributive property. The number must be distributed to each term of the polynomial. In $2\left(x+7\right)$ we can distribute $2$ to obtain the expression $2x+14$. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

### How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression

1. Multiply each term of the first polynomial by each term of the second.
2. Combine like terms.
3. Simplify.

### Example: Multiplying Polynomials Using the Distributive Property

Find the product.

$\left(2x+1\right)\left(3{x}^{2}-x+4\right)$

$\begin{array}{cc}2x\left(3{x}^{2}-x+4\right)+1\left(3{x}^{2}-x+4\right) \hfill & \text{Use the distributive property}.\hfill \\ \left(6{x}^{3}-2{x}^{2}+8x\right)+\left(3{x}^{2}-x+4\right)\hfill & \text{Multiply}.\hfill \\ 6{x}^{3}+\left(-2{x}^{2}+3{x}^{2}\right)+\left(8x-x\right)+4\hfill & \text{Combine like terms}.\hfill \\ 6{x}^{3}+{x}^{2}+7x+4 \hfill & \text{Simplify}.\hfill \end{array}$

#### Analysis of the Solution

We can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.
 $3{x}^{2}$ $-x$ $+4$ $2x$ $6{x}^{3}\\$ $-2{x}^{2}$ $8x$ $+1$ $3{x}^{2}$ $-x$ $4$

### Using the distributive property

When multiplying a binomial and another polynomial with two or more terms, be sure to multiply each term in the first to each term in the second.

### Try It

Find the product.

$\left(3x+2\right)\left({x}^{3}-4{x}^{2}+7\right)$

#### Analysis of the Solution

We can use a table to keep track of our work as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.
 $3{x}^{2}$ $-x$ $+4$ $2x$ $6{x}^{3}\\$ $-2{x}^{2}$ $8x$ $+1$ $3{x}^{2}$ $-x$ $4$

[ohm_question]3864[/ohm_question]
Watch this video to see more examples of how to use the distributive property to multiply polynomials. https://youtu.be/bwTmApTV_8o

## Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial. The FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.

### How To: Given two binomials, Multiplying Using FOIL

1. Multiply the first terms of each binomial.
2. Multiply the outer terms of the binomials.
3. Multiply the inner terms of the binomials.
4. Multiply the last terms of each binomial.
6. Combine like terms and simplify.

### Example: Using FOIL to Multiply Binomials

Use FOIL to find the product. [latex-display]\left(2x-18\right)\left(3x + 3\right)[/latex-display]

Answer: Find the product of the first terms. Find the product of the outer terms. Find the product of the inner terms. Find the product of the last terms.

$\begin{array}{cc}6{x}^{2}+6x - 54x - 54\hfill & \text{Add the products}.\hfill \\ 6{x}^{2}+\left(6x - 54x\right)-54\hfill & \text{Combine like terms}.\hfill \\ 6{x}^{2}-48x - 54\hfill & \text{Simplify}.\hfill \end{array}$

### Try It

Use FOIL to find the product. [latex-display]\left(x+7\right)\left(3x - 5\right)[/latex-display]

[ohm_question]93539[/ohm_question]