Section 5.45.4 Dividing Polynomials

Corequisite Skills review (optional warm-up)

Learning Objectives

Dividing polynomials using long division (IA 5.4.3)
Dividing polynomials using synthetic division (IA 5.4.4)

Objective 1: Dividing polynomials using long division (IA 5.4.3)

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So, let’s look carefully at the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

Warm-up Example 1

This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder.

Practice Makes Perfect

Vocabulary of the example.

Fill in the blanks.

When dividing 69 by 4,

the dividend is_________,

the divisor is________,

the quotient is________,

and the remainder is________.

We check division by multiplying the quotient by the divisor and adding the remainder.

Warm-up Example 2

Find the quotient: $({x}^{2}+9x+20)÷(x+5).$

Table 1

Write it as a long division problem. Be sure the dividend is written in descending order of powers, with no missing terms.
Divide ${x}^{2}$ by $x.$ It may help to ask yourself, “What do I need to multiply $x$ by to get ${x}^{2}$ ?”
Put the answer, $x,$ in the quotient over the $x$ term. Multiply $x$ times $x+5.$ Line up the like terms under the dividend.
Subtract ${x}^{2}+5x$ from ${x}^{2}+9x.$ You may find it easier to change the signs and then add. Then bring down the last term, 20.
Divide $4x$ by $x.$ It may help to ask yourself, “What do I need to multiply $x$ by to get $4x$ ?” Put the answer, $4$ , in the quotient over the constant term.
Multiply 4 times $x+5.$
Subtract $4x+20$ from $4x+20.$
Check: Multiply the quotient by the divisor. $\,(x+4)(x+5)$ You should get the dividend. $\,{x}^{2}+9x+20✓$

Practice Makes Perfect

Divide using long division of polynomials: $({x}^{2}+10x+21)÷(x+3)$

Sometimes division of polynomials, just like division of numbers, leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Also, if you look back at the dividends in previous examples, you will notice that the terms were written in descending order of degrees, and there were no missing degrees.

Warm-up Example 3

$(5x+{x}^{4}-{x}^{2}-6)÷(x+2)$

Notice, this polynomial is not in descending order and it is missing ${x}^{3}$ term. We need to write it in the correct order and add $0{x}^{3}$ as a placeholder.

This image illustrates the step-by-step process of polynomial long division, dividing x^4 - x^2 + 5x - 6 by x + 2, with a helpful tip on subtracting terms. — Figure 1

To check, multiply divisor by the quotient and add remainder $(x+2)({x}^{3}-2{x}^{2}+3x-1)-4$

The result should be ${x}^{4}-{x}^{2}+5x-6$

Practice Makes Perfect

Dividing polynomials using long division.

$(7x+{x}^{4}-7{x}^{2}+6)÷(x+3)$

$({x}^{3}-8)÷(x-2)$

Objective 2: Dividing polynomials using synthetic division (IA 5.4.4)

As you probably noticed, long division can be tedious. Synthetic division uses the patterns from long division as a basis to make a process much simpler by leaving the variable terms out. The same example in synthetic division format is shown next.

An image comparing polynomial long division of (x^2 + 9x + 20) by (x + 5) with the equivalent synthetic division, showing how the coefficients and results correspond. The final polynomial quotient is x + 4 with a remainder of 0. — Figure 2

Synthetic division only works when the divisor is of the form (x−c).

Warm-up Example 4

Use synthetic division to find the quotient and remainder when ${x}^{4}-16{x}^{2}+3x+12$ is divided by x+4. Note that the divisor is in the form x-(-4), so use c as the divisor.

The polynomial ${x}^{4}-16{x}^{2}+3x+12$ has its term in order with descending degree but we notice there is no x3term. We will add a 0 as a placeholder for the ${x}^{3}$ term.

An example of synthetic division for a polynomial. The divisor is -4, and the coefficients of the dividend are 1, 0, -16, 3, 12. The result shows quotient coefficients 1, -4, 0, 3 and a remainder of 0. — Figure 3

We divided a 4th degree polynomial by a 1st degree polynomial so the quotient will be a 3rd degree polynomial. Reading from the third row, the quotient has the coefficients 1,−4,0, and 3, which is ${x}^{3}-4{x}^{2}+0x+3$ . The remainder is 0.

Practice Makes Perfect

Dividing polynomials using synthetic division.

Let $f(x)={x}^{4}-5{x}^{2}+4x+12$

ⓐ Find $f(2)$
ⓑ Divide $f(x)$ by $(x-2)$ . What is the quotient? What is the remainder?

Let $f(x)={x}^{3}+2{x}^{2}-5x-6$

ⓐ Find $f(-3)$
ⓑ Divide $f(x)$ by $(x+3)$ . What is the quotient? What is the remainder?

What is the connection between $f(c)$ and the remainder when $f(x)$ is divided by $(x-c)$ ? Summarize your findings.

The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m. (National Park Service. "Lincoln Memorial Building Statistics." http://www.nps.gov/linc/historyculture/lincoln-memorial-building-statistics.htm. Accessed 4/3/2014) We can easily find the volume using elementary geometry.

\[\begin{array}{lll}V & = & l\cdot w\cdot h \\ & = & 61.5\cdot 40\cdot 30 \\ & = & 73,800\end{array}\]

So the volume is 73,800 cubic meters $(\text{m}³).$ Suppose we knew the volume, length, and width. We could divide to find the height.

\[\begin{array}{lll}h & = & \frac{V}{l\cdot w} \\ & = & \frac{73,800}{61.5\cdot 40} \\ & = & 30\end{array}\]

As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial $3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.$ The length of the solid is given by $3x;$ the width is given by $x-2.$ To find the height of the solid, we can use polynomial division, which is the focus of this section.

Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.

Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.

\[\begin{array}{lll}\text{dividend} & = & (\text{divisor}\cdot \text{quotient) + remainder} \\ 178 & = & (3\cdot 59)+1 \\ & = & 177+1 \\ & = & 178\end{array}\]

We call this the Division Algorithmand will discuss it more formally after looking at an example.

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide $2{x}^{3}-3{x}^{2}+4x+5$ by $x+2$ using the long division algorithm, it would look like this:

A step-by-step guide demonstrating polynomial long division, illustrating each calculation and corresponding operation from setting up the problem to the final subtraction.

We have found

\[\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\frac{31}{x+2}\]

\[2{x}^{3}-3{x}^{2}+4x+5=(x+2)(2{x}^{2}-7x+18)-31\]

We can identify the dividend, the divisor, the quotient, and the remainder.

Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.

Writing the result in this manner illustrates the Division Algorithm.

The Division Algorithm

The Division Algorithm states that, given a polynomial dividend $f(x)$ and a non-zero polynomial divisor $d(x)$ where the degree of $d(x)$ is less than or equal to the degree of $f(x)$ , there exist unique polynomials $q(x)$ and $r(x)$ such that

\[f(x)=d(x)q(x)+r(x)\]

$q(x)$ is the quotient and $r(x)$ is the remainder. The remainder is either equal to zero or has degree strictly less than $d(x).$

If $r(x)=0,$ then $d(x)$ divides evenly into $f(x).$ This means that, in this case, both $d(x)$ and $q(x)$ are factors of $f(x).$

How To

Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.

Set up the division problem.
Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
Multiply the answer by the divisor and write it below the like terms of the dividend.
Subtract the bottom binomial from the top binomial.
Bring down the next term of the dividend.
Repeat steps 2–5 until reaching the last term of the dividend.
If the remainder is non-zero, express as a fraction using the divisor as the denominator.

Example 1

Divide $5{x}^{2}+3x-2$ by $x+1.$

Divide the leading term of the dividend by the leading term of the divisor to start the quotient, then multiply and subtract as with long division.

This image illustrates the step-by-step process of polynomial long division, dividing (5x^2 + 3x - 2) by (x + 1) to arrive at a quotient of (5x - 2) with a remainder of 0.

The quotient is $5x-2.$ The remainder is 0. We write the result as

\[\frac{5{x}^{2}+3x-2}{x+1}=5x-2\]

\[5{x}^{2}+3x-2=(x+1)(5x-2)\]

Analysis

This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.

Example 2

Divide $6{x}^{3}+11{x}^{2}-31x+15$ by $3x-2.$

Same long-division steps as before, just note the divisor's leading coefficient isn't 1 this time.

An image demonstrating polynomial long division, showing the step-by-step calculation of (6x^3 + 11x^2 - 31x + 15) divided by (3x - 2) which results in a quotient of (2x^2 + 5x - 7) and a remainder of 1. Each mathematical step is explained alongside it.

There is a remainder of 1. We can express the result as:

\[\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x-2}=2{x}^{2}+5x-7+\frac{1}{3x-2}\]

Analysis

We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.

\[(3x-2)(2{x}^{2}+5x-7)+1=6{x}^{3}+11{x}^{2}-31x+15\]

Notice, as we write our result,

the dividend is $6{x}^{3}+11{x}^{2}-31x+15$
the divisor is $3x-2$
the quotient is $2{x}^{2}+5x-7$
the remainder is $1$

Try It #1

Divide $16{x}^{3}-12{x}^{2}+20x-3$ by $4x+5.$

$4{x}^{2}-8x+15-\frac{78}{4x+5}$

Did you get it?

Using Synthetic Division to Divide Polynomials

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 linear factor whose leading coefficient is 1.

To illustrate the process, recall the example at the beginning of the section.

Divide $2{x}^{3}-3{x}^{2}+4x+5$ by $x+2$ using the long division algorithm.

The final form of the process looked like this:

A polynomial long division problem showing (2x^3 - 3x^2 + 4x + 5) divided by (x + 2), resulting in a quotient of (2x^2 - 7x + 18) and a remainder of -31.

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

Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.

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 $2{x}^{2}-7x+18$ and the remainder is $-31.$ The process will be made more clear in Example 3.

Synthetic Division

Synthetic division is a shortcut that can be used when the divisor is a binomial in the form $x-k$ where $k$ is a real number. In synthetic division, only the coefficients are used in the division process.

How To

Given two polynomials, use synthetic division to divide.

Write $k$ for the divisor.
Write the coefficients of the dividend.
Bring the lead coefficient down.
Multiply the lead coefficient by $k.$ Write the product in the next column.
Add the terms of the second column.
Multiply the result by $k.$ Write the product in the next column.
Repeat steps 5 and 6 for the remaining columns.
Use the bottom numbers to write the quotient. The number in the last column is the remainder. The next number from the right has degree 0, the next number has degree 1, and so on.

Example 3

Use synthetic division to divide $5{x}^{2}-3x-36$ by $x-3.$

Set $k=3$ (from $x-3$), bring down the leading coefficient, then multiply-and-add across each column.

Begin by setting up the synthetic division. Write $k$ and the coefficients.

A collapsed version of the previous synthetic division.

Bring down the lead coefficient. Multiply the lead coefficient by $k.$

$The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.$

Continue by adding the numbers in the second column. Multiply the resulting number by $k.$ Write the result in the next column. Then add the numbers in the third column.

Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.

The result is $5x+12.$ The remainder is 0. So $x-3$ is a factor of the original polynomial.

Analysis

Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.

$(x-3)(5x+12)+0=5{x}^{2}-3x-36$

Example 4

Use synthetic division to divide $4{x}^{3}+10{x}^{2}-6x-20$ by $x+2.$

Since the divisor is $x+2$, use $k=-2$ in the synthetic division.

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

Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.

The result is $4{x}^{2}+2x-10.$ The remainder is 0. Thus, $x+2$ is a factor of $4{x}^{3}+10{x}^{2}-6x-20.$

Analysis

The graph of the polynomial function $f(x)=4{x}^{3}+10{x}^{2}-6x-20$ in Figure 5 shows a zero at $x=k=-2.$ This confirms that $x+2$ is a factor of $4{x}^{3}+10{x}^{2}-6x-20.$

Example 5

Use synthetic division to divide $-9{x}^{4}+10{x}^{3}+7{x}^{2}-6$ by $x-1.$

Insert a 0 coefficient for the missing x-term before setting up the synthetic division.

Notice there is no x-term. We will use a zero as the coefficient for that term.

A synthetic division problem is shown, with the divisor 1. The dividend's coefficients are -9, 10, 7, 0, -6. The result of the division is a quotient with coefficients -9, 1, 8, 8 and a remainder of 2.

The result is $-9{x}^{3}+{x}^{2}+8x+8+\frac{2}{x-1}.$

Try It #2

Use synthetic division to divide $3{x}^{4}+18{x}^{3}-3x+40$ by $x+7.$

$3{x}^{3}-3{x}^{2}+21x-150+\frac{1,090}{x+7}$

Did you get it?

Using Polynomial Division to Solve Application Problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.

Example 6

The volume of a rectangular solid is given by the polynomial $3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.$ The length of the solid is given by $3x$ and the width is given by $x-2.$ Find the height, $h,$ of the solid.

Substitute the known volume, length, and width into $V=l\cdot w\cdot h$, then divide out the length and width to solve for the height.

There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in Figure 6.

Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2. — Figure 6

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

\[\begin{array}{lll}V & = & l\cdot w\cdot h \\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x & = & 3x\cdot (x-2)\cdot h\end{array}\]

To solve for $h,$ first divide both sides by $3x.$

\[\begin{array}{lll}\frac{3x\cdot (x-2)\cdot h}{3x} & = & \frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\ (x-2)h & = & {x}^{3}-{x}^{2}-11x+18\end{array}\]

Now solve for $h$ using synthetic division.

\[h=\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}\]

Synthetic division for the polynomial x^3 - x^2 - 11x + 18 divided by x - 2, showing a quotient of x^2 + x - 9 and a remainder of 0.

The quotient is ${x}^{2}+x-9$ and the remainder is 0. The height of the solid is ${x}^{2}+x-9.$

Try It #3

The area of a rectangle is given by $3{x}^{3}+14{x}^{2}-23x+6.$ The width of the rectangle is given by $x+6.$ Find an expression for the length of the rectangle.

$3{x}^{2}-4x+1$

Did you get it?

Media

Access these online resources for additional instruction and practice with polynomial division.

Key Equations

Table 2
Division Algorithm	$f(x)=d(x)q(x)+r(x)\,\text{where }q(x)\ne 0$

Key Concepts

Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See Example 1 and Example 2.
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form $x-k.$ See Example 3,Example 4,and Example 5.
Polynomial division can be used to solve application problems, including area and volume. See Example 6.

Section Exercises

Verbal

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

The binomial is a factor of the polynomial.

If a polynomial of degree $n$ is divided by a binomial of degree 1, what is the degree of the quotient?

Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$({x}^{2}+5x-1)÷(x-1)$

$x+6+\frac{5}{x-1}\text{,}\,\text{quotient:}\,x+6\text{,}\,\text{remainder:}\,\text{5}$

$(2{x}^{2}-9x-5)÷(x-5)$

$(3{x}^{2}+23x+14)÷(x+7)$

$3x+2\text{,}\,\text{quotient: }3x+2\text{,}\,\text{remainder: 0}$

$(4{x}^{2}-10x+6)÷(4x+2)$

$(6{x}^{2}-25x-25)÷(6x+5)$

$x-5\text{,}\,\text{quotient:}\,x-5\text{,}\,\text{remainder:}\,\text{0}$

$(-{x}^{2}-1)÷(x+1)$

$(2{x}^{2}-3x+2)÷(x+2)$

$2x-7+\frac{16}{x+2}\text{,}\,\text{quotient:}\,\,2x-7\text{,}\,\text{remainder:}\,\text{16}$

$({x}^{3}-126)÷(x-5)$

$(3{x}^{2}-5x+4)÷(3x+1)$

$x-2+\frac{6}{3x+1}\text{,}\,\text{quotient:}\,x-2\text{,}\,\text{remainder:}\,\text{6}$

$({x}^{3}-3{x}^{2}+5x-6)÷(x-2)$

$(2{x}^{3}+3{x}^{2}-4x+15)÷(x+3)$

$2{x}^{2}-3x+5\text{,}\,\text{quotient:}\,2{x}^{2}-3x+5\text{,}\,\text{remainder:}\,\text{0}$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$(3{x}^{3}+2{x}^{2}-x+4)÷(x-3)$

$(2{x}^{3}-6{x}^{2}-7x+6)÷(x-4)$

$2{x}^{2}+2x+1+\frac{10}{x-4}$

$(6{x}^{3}-10{x}^{2}-7x-15)÷(x+1)$

$(4{x}^{3}-12{x}^{2}-5x-1)÷(2x+1)$

$2{x}^{2}-7x+1-\frac{2}{2x+1}$

$(9{x}^{3}-9{x}^{2}+18x+5)÷(3x-1)$

$(3{x}^{3}-2{x}^{2}+x-4)÷(x+3)$

$3{x}^{2}-11x+34-\frac{106}{x+3}$

$(-6{x}^{3}+{x}^{2}-4)÷(2x-3)$

$(2{x}^{3}+7{x}^{2}-13x-3)÷(2x-3)$

${x}^{2}+5x+1$

$(3{x}^{3}-5{x}^{2}+2x+3)÷(x+2)$

$(4{x}^{3}-5{x}^{2}+13)÷(x+4)$

$4{x}^{2}-21x+84-\frac{323}{x+4}$

$({x}^{3}-3x+2)÷(x+2)$

$({x}^{3}-21{x}^{2}+147x-343)÷(x-7)$

${x}^{2}-14x+49$

$({x}^{3}-15{x}^{2}+75x-125)÷(x-5)$

$(9{x}^{3}-x+2)÷(3x-1)$

$3{x}^{2}+x+\frac{2}{3x-1}$

$(6{x}^{3}-{x}^{2}+5x+2)÷(3x+1)$

$({x}^{4}+{x}^{3}-3{x}^{2}-2x+1)÷(x+1)$

${x}^{3}-3x+1$

$({x}^{4}-3{x}^{2}+1)÷(x-1)$

$({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6)÷(x+3)$

${x}^{3}-{x}^{2}+2$

$({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36)÷(x-2)$

$({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16)÷(x-2)$

${x}^{3}-6{x}^{2}+12x-8$

$({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10)÷(x+5)$

$({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81)÷(x-3)$

${x}^{3}-9{x}^{2}+27x-27$

$(4{x}^{4}-2{x}^{3}-4x+2)÷(2x-1)$

$(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2)÷(2x+1)$

$2{x}^{3}-2x+2$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x-2,\,4{x}^{3}-3{x}^{2}-8x+4$

$x-2,\,3{x}^{4}-6{x}^{3}-5x+10$

Yes $(x-2)(3{x}^{3}-5)$

$x+3,\,-4{x}^{3}+5{x}^{2}+8$

$x-2,\,4{x}^{4}-15{x}^{2}-4$

Yes $(x-2)(4{x}^{3}+8{x}^{2}+x+2)$

$x-\frac{1}{2},\,2{x}^{4}-{x}^{3}+2x-1$

$x+\frac{1}{3},\,3{x}^{4}+{x}^{3}-3x+1$

Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is ${x}^{2}-x+3$

Graph of a polynomial that has a x-intercept at -1.

Factor is $({x}^{2}+2x+4)$

$(x-1)({x}^{2}+2x+4)$

Factor is ${x}^{2}+2x+5$

Graph of a polynomial that has a x-intercept at 2.

Factor is ${x}^{2}+x+1$

Graph of a polynomial that has a x-intercept at 5.

$(x-5)({x}^{2}+x+1)$

Factor is ${x}^{2}+2x+2$

Graph of a polynomial that has a x-intercept at -3.

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{4{x}^{3}-33}{x-2}$

$\text{Quotient:}\,4{x}^{2}+8x+16\text{,}\,\text{remainder:}\,-1$

$\frac{2{x}^{3}+25}{x+3}$

$\frac{3{x}^{3}+2x-5}{x-1}$

$\text{Quotient:}\,3{x}^{2}+3x+5\text{,}\,\text{remainder:}\,0$

$\frac{-4{x}^{3}-{x}^{2}-12}{x+4}$

$\frac{{x}^{4}-22}{x+2}$

$\text{Quotient:}\,{x}^{3}-2{x}^{2}+4x-8\text{,}\,\text{remainder:}\,-6$

Technology

For the following exercises, use a calculator with CAS to answer the questions.

Consider $\frac{{x}^{k}-1}{x-1}$ with $k=1, 2, 3.$ What do you expect the result to be if $k=4?$

Consider $\frac{{x}^{k}+1}{x+1}$ for $k=1, 3, 5.$ What do you expect the result to be if $k=7?$

${x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1$

Consider $\frac{{x}^{4}-{k}^{4}}{x-k}$ for $k=1, 2, 3.$ What do you expect the result to be if $k=4?$

Consider $\frac{{x}^{k}}{x+1}$ with $k=1, 2, 3.$ What do you expect the result to be if $k=4?$

${x}^{3}-{x}^{2}+x-1+\frac{1}{x+1}$

Consider $\frac{{x}^{k}}{x-1}$ with $k=1, 2, 3.$ What do you expect the result to be if $k=4?$

Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x+1}{x-i}$

$1+\frac{1+i}{x-i}$

$\frac{{x}^{2}+1}{x-i}$

$\frac{x+1}{x+i}$

$1+\frac{1-i}{x+i}$

$\frac{{x}^{2}+1}{x+i}$

$\frac{{x}^{3}+1}{x-i}$

${x}^{2}+ix-1+\frac{1-i}{x-i}$

Real-World Applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

Length is $x+5,$ area is $2{x}^{2}+9x-5.$

Length is $2x\,+\,5,$ area is $4{x}^{3}+10{x}^{2}+6x+15$

$2{x}^{2}+3$

Length is $3x-4,$ area is $6{x}^{4}-8{x}^{3}+9{x}^{2}-9x-4$

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $12{x}^{3}+20{x}^{2}-21x-36,$ length is $2x+3,$ width is $3x-4.$

$2x+3$

Volume is $18{x}^{3}-21{x}^{2}-40x+48,$ length is $3x-4,$ width is $3x-4.$

Volume is $10{x}^{3}+27{x}^{2}+2x-24,$ length is $5x-4,$ width is $2x+3.$

$x+2$

Volume is $10{x}^{3}+30{x}^{2}-8x-24,$ length is $2,$ width is $x+3.$

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $\pi (25{x}^{3}-65{x}^{2}-29x-3),$ radius is $5x+1.$

$x-3$

Volume is $\pi (4{x}^{3}+12{x}^{2}-15x-50),$ radius is $2x+5.$

Volume is $\pi (3{x}^{4}+24{x}^{3}+46{x}^{2}-16x-32),$ radius is $x+4.$

$3{x}^{2}-2$

Glossary

Division Algorithm: given a polynomial dividend $f(x)$ and a non-zero polynomial divisor $d(x)$ where the degree of $d(x)$ is less than or equal to the degree of $f(x)$ , there exist unique polynomials $q(x)$ and $r(x)$ such that $f(x)=d(x)q(x)+r(x)$ where $q(x)$ is the quotient and $r(x)$ is the remainder. The remainder is either equal to zero or has degree strictly less than $d(x).$
synthetic division: a shortcut method that can be used to divide a polynomial by a binomial of the form $x-k$


Write it as a long division problem. Be sure the dividend is written in descending order of powers, with no missing terms.
Divide \({x}^{2}\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \({x}^{2}\) ?”
Put the answer, \(x,\) in the quotient over the \(x\) term. Multiply \(x\) times \(x+5.\) Line up the like terms under the dividend.
Subtract \({x}^{2}+5x\) from \({x}^{2}+9x.\) You may find it easier to change the signs and then add. Then bring down the last term, 20.
Divide \(4x\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \(4x\) ?” Put the answer, \(4\) , in the quotient over the constant term.
Multiply 4 times \(x+5.\)
Subtract \(4x+20\) from \(4x+20.\)
Check: Multiply the quotient by the divisor. \(\,(x+4)(x+5)\) You should get the dividend. \(\,{x}^{2}+9x+20✓\)