In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a polynomial.

A polynomial function is a function consisting of sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variables with an non-negative integer exponents. Non negative integers are 0, 1, 2, 3, 4, …
You may see a resemblance between expressions and polynomials, which we have been studying in this course.  Polynomials are a special sub-group of mathematical expressions and equations.

The following table is intended to help you tell the difference between what is a polynomial and what is not.

The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.

A polynomial containing two terms, such as [latex]2x - 9[/latex], is called a binomial. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[/latex], 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. It is important to note that polynomials only have integer exponents.

In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial.

The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.

When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn’t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[/latex].

A term without a variable is called a constant term, and the degree of that term is 0. For example 13 is the constant term in [latex]3y+13[/latex]. You would usually say that [latex]14y^{3}+3y[/latex] has no constant term or that the constant term is 0.

Evaluate a polynomial

You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.

 IN the following video we show more examples of evaluating polynomials for given values of the variable.

Add and Subtract Polynomials

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, [latex]5{x}^{2}[/latex] and [latex]-2{x}^{2}[/latex] are like terms, and can be added to get [latex]3{x}^{2}[/latex], but [latex]3x[/latex] and [latex]3{x}^{2}[/latex] are not like terms, and therefore cannot be added.

Here is a summary of some helpful steps for adding and subtracting polynomials.

When you subtract polynomials you will still be looking for like terms to combine, but you will need to pay attention to the sign of the terms you are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.

Example

Find the difference.

[latex]\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)[/latex]

Show Answer

[latex]\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]

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.

In the following video we show more examples of adding and subtracting polynomials.

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.

You may have used the distributive property to help you solve linear equations such as [latex]2\left(x+7\right)=21[/latex]. We can distribute the [latex]2[/latex] in [latex]2\left(x+7\right)[/latex] to obtain the equivalent expression [latex]2x+14[/latex]. 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.

The following video will provide you with examples of using the distributive property to find the product of monomials and polynomials.

Below is a summary of the steps we used to find the product of two polynomials using the distributive property.

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.

Using FOIL to Multiply Binomials

We can also use a shortcut called the FOIL method when multiplying 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 arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

Example

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

Show 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.

[latex]\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}[/latex]

In this video, we show an example of how to use the FOIL method to multiply two binomials.

The following steps summarize the process for using FOIL to multiply two binomials.  It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial and a trinomial, it is better to use a table to keep track of your terms.

How To: Given two binomials, use FOIL to simplify the expression.

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.
5. Add the products.
6. Combine like terms and simplify.

Multiply a Trinomial and a Binomial

Another type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method can not be used since there are more than two terms in a trinomial, you still use the Distributive Property to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial.

For our first examples, we will show you two ways to organize all of the terms that result from multiplying polynomials with more than two terms. The most important part of the process is finding a way to organize terms.

As you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problem as above, we will show another way to organize all the terms produced by multiplying two polynomials with more than two terms.

Notice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the Distributive Property to multiply a binomial by a trinomial.

In our next example we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms.  Forgetting a negative sign is the easiest mistake to make in this case.

In the following video we show more examples of multiplying polynomials.

Divide a polynomial by a binomial

Dividing a polynomial by a monomial can be handled by dividing each term in the polynomial separately. This can’t be done when the divisor has more than one term. However, the process of long division can be very helpful with polynomials.

Recall how you can use long division to divide two whole numbers, say 900 divided by 37.

[latex]37\overline{)900}[/latex]

First, you would think about how many 37s are in 90, as 9 is too small. (Note: you could also think, how many 40s are there in 90.)

[latex]\begin{array}\,\,\,\,\,\,\,\,\,\,\,\,2\\37\overline{)900}\\\,\,\,\,\,\,\,\,74\end{array}[/latex]

There are two 37s in 90, so write 2 above the last digit of 90. Two 37s is 74; write that product below the 90.

[latex]\begin{array}\,\,\,\,\,\,\,\,\,\,\,\,2\\37\overline{)900}\\\,\,\,\,\,\underline{-74}\\\,\,\,\,\,\,\,16\end{array}[/latex]

Subtract: [latex]90–74[/latex] is 16. (If the result is larger than the divisor, 37, then you need to use a larger number for the quotient.)

[latex]\begin{array}\,\,\,\,\,\,\,\,\,\,\,\,2\\37\overline{)900}\\\,\,\,\,\,\underline{-74}\\\,\,\,\,\,\,\,160\end{array}[/latex]

Bring down the next digit (0) and consider how many 37s are in 160.

[latex]\begin{array}\,\,\,\,\,\,\,\,\,\,\,\,2\\37\overline{)900}\\\,\,\,\,\,\underline{-74}\\\,\,\,\,\,\,\,160\\\,\,\,\underline{-148}\end{array}[/latex]

There are four 37s in 160, so write the 4 next to the two in the quotient. Four 37s is 148; write that product below the 160.

Subtract: [latex]160–148[/latex] is 12. This is less than 37 so the 4 is correct. Since there are no more digits in the dividend to bring down, you’re done.

The final answer is 24 R12, or [latex]24\frac{12}{37}[/latex]. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:

[latex]24\cdot37+12=888+12=900[/latex]

To divide polynomials, use the same process. This example shows how to do this when dividing by a binomial.

Example

Divide: [latex]\frac{\left(x^{2}–4x–12\right)}{\left(x+2\right)}[/latex]

Show Solution

How many x’s are there in [latex]x^{2}[/latex]? That is, what is [latex] \frac{{{x}^{2}}}{x}[/latex]?

[latex] \frac{{{x}^{2}}}{x}=x[/latex]. Put x in the quotient above the [latex]-4x[/latex] term. (These are like terms, which helps to organize the problem.)

Write the product of the divisor and the part of the quotient you just found under the dividend. Since [latex]x\left(x+2\right)=x^{2}+2x[/latex], write this underneath, and get ready to subtract.

Rewrite [latex]–\left(x^{2} + 2x\right)[/latex] as its opposite [latex]–x^{2}–2x[/latex] so that you can add the opposite. (Adding the opposite is the same as subtracting, and it is easier to do.)

Add [latex]-x^{2}[/latex] to [latex]x^{2}[/latex], and [latex]-2x[/latex] to [latex]-4x[/latex].

Bring down [latex]-12[/latex].

Repeat the process. How many times does x go into [latex]-6x[/latex]? In other words, what is [latex] \frac{-6x}{x}[/latex]?

Since [latex] \frac{-6x}{x}=-6[/latex], write [latex]-6[/latex] in the quotient. Multiply [latex]-6[/latex] and [latex]x+2[/latex] and prepare to subtract the product.

Rewrite [latex]–\left(-6x–12\right)[/latex] as [latex]6x+12[/latex], so that you can add the opposite.

Add. In this case, there is no remainder, so you’re done.

Answer

[latex]\frac{\left(x^{2}–4x–12\right)}{\left(x+2\right)}=x–6[/latex]

Check this by multiplying:

[latex]\left(x-6\right)\left(x+2\right)=x^{2}+2x-6x-12=x^{2}-4x-12[/latex]

In this video we show another example of dividing a degree two trinomial by a degree one binomial.

Let’s try another example. In this example, a term is “missing” from the dividend.

Example

Divide: [latex]\frac{\left(x^{3}–6x–10\right)}{\left(x–3\right)}[/latex]

Show Solution

In setting up this problem, notice that there is an [latex]x^{3}[/latex] term but no [latex]x^{2}[/latex] term. Add [latex]0x^{2}[/latex] as a “place holder” for this term. (Since 0 times anything is 0, you’re not changing the value of the dividend.)

Focus on the first terms again: how many x’s are there in [latex]x^{3}[/latex]? Since [latex] \frac{{{x}^{3}}}{x}=x^{2}[/latex], put [latex]x^{2}[/latex] in the quotient.

Multiply [latex]x^{2}\left(x–3\right)=x^{3}–3x^{2}[/latex], write this underneath the dividend, and prepare to subtract.

Rewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[/latex]. Then add.

Bring down the rest of the expression in the dividend. It’s helpful to bring down all of the remaining terms.

Now, repeat the process with the remaining expression, [latex]3x^{2}-6x–10[/latex], as the dividend.

Remember to watch the signs!

How many x’s are there in 3x? Since there are 3, multiply [latex]3\left(x–3\right)=3x–9[/latex], write this underneath the dividend, and prepare to subtract.

Continue until the degree of the remainder is less than the degree of the divisor. In this case the degree of the remainder, [latex]-1[/latex], is 0, which is less than the degree of [latex]x-3[/latex], which is 1.

Also notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.

You can write the remainder using the symbol R, or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.

Answer

[latex]\begin{array}{r}{\left(x^{3}–6x–10\right)}{\left(x–3\right)}=x^{2}+3x+3+R-1,\\x^{2}+3x+3+\frac{-1}{x-3}, \text{ or }\\x^{2}+3x+3-\frac{1}{x-3}\end{array}[/latex]

Check the result:

[latex]\left(x–3\right)\left(x^{2}+3x+3\right)\,\,\,=\,\,\,x\left(x^{2}+3x+3\right)–3\left(x^{2}+3x+3\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,x^{3}+3x^{2}+3x–3x^{2}–9x–9\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,x^{3}–6x–9\\\,\,\,\,\,\,\,\,x^{3}–6x–9+\left(-1\right)\,\,\,=\,\,\,x^{3}–6x–10[/latex]

In the video that follows, we show another example of dividing a degree three trinomial by a binomial, not the “missing” term and how we work with it.

The process above works for dividing any polynomials, no matter how many terms are in the divisor or the dividend. The main things to remember are:

• When subtracting, be sure to subtract the whole expression, not just the first term. This is very easy to forget, so be careful!
• Stop when the degree of the remainder is less than the degree of the dividend, or when you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend.

In this video we present one more example of polynomial long division.

Synthetic Division

As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a polynomial whose leading coefficient is 1.

To illustrate the process, divide [latex]2{x}^{3}-3{x}^{2}+4x+5[/latex] by [latex]x+2[/latex] using the long division algorithm.

There is a lot of repetition in this process. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign, we already have a simpler version of the entire problem.

Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.

We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[/latex] and the remainder is –31. The process will be made more clear in the following example.

Analysis of the solution

It is important to note that the result, [latex]5x+12[/latex], of [latex]5{x}^{2}-3x - 36\div{x-3}[/latex] is one degree less than[latex]5{x}^{2}-3x - 36[/latex]. Why is that? Think about how you would have solved this using long division. The first thing you would ask yourself is how many x’s are in [latex]5x^2[/latex]?

[latex]x-3\overline{)5{x}^{2}-3x - 36}[/latex]

To get a result of [latex]5x^2[/latex], you need to multiply [latex]x[/latex] by [latex]5x[/latex].  The next step in long division is to subtract this result from [latex]5x^2[/latex].  This leaves us with no [latex]x^2[/latex] term in the result.

In this video example, you will see another example of using synthetic division for division of a degree two polynomial by a degree one binomial.

In the next example we will use synthetic division to divide a third-degree polynomial.

Example

Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[/latex] by [latex]x+2[/latex].

Show Answer

The binomial divisor is [latex]x+2[/latex] so [latex]k=-2[/latex]. Add each column, multiply the result by –2, and repeat until the last column is reached.

The result is [latex]4{x}^{2}+2x - 10[/latex]. Again notice the degree of the result is less than the degree of the quotient, [latex]4{x}^{3}+10{x}^{2}-6x - 20[/latex].

We can check that we are correct by multiplying the result with the divisor:

[latex](x+2)(4{x}^{2}+2x - 10)=4x^3+2x^2-10x+8x^2+4x-20=4x^3+10x^2-6x-20[/latex]

Answer

[latex]4{x}^{3}+10{x}^{2}-6x - 20\div{x+2}=4{x}^{2}+2x - 10[/latex]

In the next example we will show division of a fourth degree polynomial by a binomial.  Note how there is no x term in the fourth degree polynomial, so we need to use a placeholder of 0 to ensure proper alignment of terms.

In our last video example we show another example of how to use synthetic division to divide a degree three polynomial by a degree one binomial.

Summary

Multiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.

Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation, or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.