Precalculus by Richard Wright

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Blessed are the merciful, for they will be shown mercy. Matthew‬ ‭5‬:‭7‬ ‭NIV‬‬‬‬

2-04 Dividing Polynomials

Solomon's Temple is described in detail in 1 Kings 6. The main sanctuary section is described as a rectangular prism with interior dimensions of 20 cubits wide, 60 cubits long, and 30 cubits high. The cubit is the distance from the elbow to the tips of the fingers on an average person. There were two different cubits in Old Testament times, the basic cubit of about 17.5 inches and the royal cubit which is about 20 inches. The longer royal cubit was likely used in this case making the interior dimensions of Solomon's Temple approximately 33 feet wide, 100 feet long, and 50 feet high. That is not very large compared to buildings today. My modest house is a rectangle about 25 feet wide by 100 feet long, or about the same footprint as the interior of the Temple! The interior volume of the Temple can be found using elementary geometry.

V = l ⋅ w ⋅ h
= (60 cubits) ⋅ (20 cubits) ⋅ (30 cubits)
= 36,000 cubits³

So the interior volume of Solomon's Temple was 36,000 cubic cubits. What if only the volume, length, and width were known? We could divide to find the height.

$$ h = \frac{V}{l·w} $$

$$ = \frac{36,000 \text{ cubits}^3}{(60 \text{ cubits})(20 \text{ cubits})} $$

$$ = 30 \text{ cubits} $$

Just like stated above, the height is 30 cubits. But what if a model maker wants to print a scale model? He could build a digital scale model where the dimensions are given using variable terms. If he wants a certain size base and a certain volume, then he could use division to find the height using division with variables. This lesson focuses on how to divide polynomials with variables.

Polynomial Long Division

Long division with integers was taught in elementary school. The long division algorithm is divide the numbers on the left, multiply the answer with the divisor, subtract, bring down the next digit, and repeat until all the digits have been done. For example, consider dividing 137 by 3.

$$ \require{enclose} \begin{array}{rll} 45 \\ 3 \enclose{longdiv}{137} && \\ \underline{−12}\downarrow && \hbox{(\(3 × 4 = 12\))} \\ 17 && \hbox{(\(13 - 12 = 1\)), Bring down the 7} \\ \underline{-15} && \hbox{(\(3 × 5 = 15\))} \\ 2 && \hbox{(\(17 - 15 = 2\))} \\ && \hbox{Answer: 45 R 2 or \(45 \tfrac{2}{3}\)} \\ \end{array}$$

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.

dividend = (divisor · quotient) + remainder
137 = (3 · 45) + 2
137 = 135 + 2
137 = 137

This is called the division algorithm and will be discussed more formally after looking at an example.

Division of polynomials is similar to long division of whole numbers. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. For example, divide 3x³ − 2x² + 5x + 1 by x + 1 using the long division algorithm, it would look like this:

Set up the division.

$$ \require{enclose} x + 1 \enclose{longdiv}{3x^3 - 2x^2 + 5x + 1} $$

3x² divided by x.

$$ \require{enclose} \begin{array}{rl} \color{blue}{3x^2} \phantom{ + 10x + 4} \\ \color{blue}{x} + 1 \enclose{longdiv}{\color{blue}{3x^3} - 2x^2 + 5x + 1} \\ \end{array} $$

3x² multiplied by x + 1.

$$ \require{enclose} \begin{array}{rl} \color{blue}{3x^2} \phantom{ + 10x + 4} \\ \color{blue}{x + 1} \enclose{longdiv}{3x^3 - 2x^2 + 5x + 1} \\ \color{blue}{3x^3 + 3x^2} \phantom{ + 10x + 4} \\ \end{array} $$

Subtract and bring down next term.

$$ \require{enclose} \begin{array}{rl} 3x^2 \phantom{ + 10x + 10} && \\ x + 1 \enclose{longdiv}{\color{blue}{3x^3 - 2x^2} + 5x + \phantom{0}1} \\ \underline{\color{blue}{-(3x^3 + 3x^2)}} \phantom{+} \downarrow \phantom{x + 4} \\ \color{blue}{-5x^2 + 5x} \phantom{0 + 4} \\ \end{array} $$

−5x² divided by x.

$$ \require{enclose} \begin{array}{rl} 3x^2 \color{blue}{- 5x} \phantom{ + 100} \\ \color{blue}{x} + 1 \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ \color{blue}{-5x^2} + 5x \phantom{0 + 4} \\ \end{array} $$

−5x multiplied by x + 1.

$$ \require{enclose} \begin{array}{rl} 3x^2 \color{blue}{- 5x} \phantom{ + 100} \\ \color{blue}{x + 1} \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ -5x^2 + 5x \phantom{0 + 4} \\ \color{blue}{-5x^2 - 5x} \phantom{0 + 4} \\ \end{array} $$

Subtract and bring down next term.

$$ \require{enclose} \begin{array}{rl} 3x^2 - 5x \phantom{ + 100} \\ x + 1 \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ \color{blue}{-5x^2 + 5x} \phantom{0 + 4} \\ \underline{\color{blue}{-(-5x^2 - 5x)}} \phantom{10} \downarrow \\ \color{blue}{10x + 1} \\ \end{array} $$

10x divided by x.

$$ \require{enclose} \begin{array}{rl} 3x^2 - 5x \color{blue}{ + 10} \\ \color{blue}{x} + 1 \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ -5x^2 + 5x \phantom{0 + 4} \\ \underline{-(-5x^2 - 5x)} \phantom{10} \downarrow \\ \color{blue}{10x} + 1 \\ \end{array} $$

10 multiplied by x + 1.

$$ \require{enclose} \begin{array}{rl} 3x^2 - 5x \color{blue}{ + 10} \\ \color{blue}{x + 1} \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ -5x^2 + 5x \phantom{0 + 4} && \\ \underline{-(-5x^2 - 5x)} \phantom{10} \downarrow \\ 10x + \phantom{0}1 \\ \color{blue}{10x + 10} \\ \end{array} $$

Subtract.

$$ \require{enclose} \begin{array}{rl} \color{red}{3x^2 - 5x + 10} \\ \color{purple}{x + 1} \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} \\ -5x^2 + 5x \phantom{0 + 4} \\ \underline{-(-5x^2 - 5x)} \phantom{10} \downarrow \\ \color{blue}{10x + \phantom{0}1} \\ \underline{\color{blue}{-(10x + 10)}} \\ \color{blue}{-9} \\ \end{array}$$

Answer: \(\color{red}{3x^2 - 5x + 10} \color{blue}{-} \frac{\color{blue}{9}}{\color{purple}{x + 1}}\)

This example found

$$ \frac{3x^3 - 2x^2 + 5x + 1}{x+ 1} = 3x^2 - 5x + 10 - \frac{9}{x + 1} $$

or by multiplying both sides by x + 1

3x³ − 2x² + 5x + 1 = (x + 1)·(3x² − 5x + 10) − 9

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

$$ \begin{array}{cccc} \color{blue}{3x^3 - 2x^2 + 5x + 1} & = \color{red}{(x + 1)} & · & \color{purple}{(3x^2 - 5x + 10)} & \color{green}{-9} \\ \color{blue}{\text{Dividend}} & \color{red}{\text{Divisor}} && \color{purple}{\text{Quotient}} & \color{green}{\text{Remainder}} \end{array} $$

Writing the result in this manner illustrates the Division Algorithm.

Use Synthetic Division to Divide Polynomials

As is evident from the previous examples, long division can be long and cumbersome. Synthetic Division is a shorthand methods of dividing polynomials by a binomial, x − k. Notice the leading coefficient of the binomial is 1.

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

Divide 3x³ − 2x² + 5x + 1 by x + 1 using long division. The final form of the process looked like this:

$$ \require{enclose} \begin{array}{rll} 3x^2 - 5x + 10 && \\ x + 1 \enclose{longdiv}{3x^3 - 2x^2 + 5x + \phantom{0}1} && \\ \underline{-(3x^3 + 3x^2)} \phantom{+} \downarrow \phantom{x + 4} && \\ -5x^2 + 5x \phantom{0 + 4} && \\ \underline{-(-5x^2 - 5x)} \phantom{10} \downarrow && \\ 10x + \phantom{0}1 && \\ \underline{-(10x + 10)} && \\ -9 && \\ \end{array}$$

There is a lot of repetition in the table. To simplify it, don’t write the variables, but line up their coefficients in columns under the division sign. Also, eliminate the partial products to create a simpler version of the entire problem.

$$ \require{enclose} \begin{array}{rll} 3 \quad -5 \quad\quad 10 && \\ 1 \enclose{longdiv}{3 \quad -2 \quad\quad 5 \quad\quad \phantom{0}1} && \\ \underline{-3 \quad -3}\quad \phantom{+} \downarrow \phantom{000}\quad && \\ \underline{5 \quad\quad 5} \phantom{10} \quad \downarrow && \\ \underline{-10 \; -10} && \\ -9 && \\ \end{array}$$

Synthetic division further simplifies even more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 1, as is done in division of whole numbers, then multiplying and subtracting the middle product, change the sign of the “divisor” to –1, multiply and add. The process starts by bringing down the leading coefficient.

$$ \begin{array}{rrrrr} \underline{-1}| & 3 & -2 & 5 & 1 \\ & & -3 & 5 & -10 \\ \hline & 3 & -5 & 10 & |\underline{-9} \end{array} $$

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 3x² − 5x + 10 and the remainder is −9. The process will be made more clear in example 4.

Example 5: Synthetic Division

Use synthetic division to divide 2x³ − x² − 26x + 33 by x − 3.

Solution

The binomial divisor is x − 3, so k = 3. Add each column, multiply the result by 3, and repeat until the last column is reached.

$$ \begin{array}{rrrrr} \underline{3}| & 2 & -1 & -26 & 33 \\ & & 6 & 15 & -33 \\ \hline & 2 & 5 & -11 & |\underline{\phantom{00}0} \end{array} $$

The result is 2x² + 5x − 11. The remainder is 0, so x − 3 is a factor of 2x³ − x² − 26x + 33.

Analysis The graph of the polynomial function f(x) = 2x³ − x² − 26x + 33 in figure 2 shows a zero at x = k = 3 because there is an x-intercept at x = 3. This confirms that x − 3 is a factor of 2x³ − x² − 26x + 33.

Use Polynomial Division to Solve Application Problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. An example of this was at the beginning of this section.

Example 7: Use Polynomial Division in an Application Problem

The volume of a rectangular solid is given by 2x³ − 4x² − 6x. The length of the solid is 2x and the width is x + 1. Find the height of the solid.

Solution

There are a few ways to approach this problem. Divide the expression for the volume of the solid by the expressions for the length and width. A sketch is in figure 3.

Write an equation by substituting the known values into the formula for the volume of a rectangular solid.

V = l · w · h

2x³ − 4x² − 6x = 2x · (x + 1) · h

To solve for h, first divide both sides by 2x.

$$ \frac{2x^3 - 4x^2 - 6x}{2x} = \frac{2x · (x + 1) · h}{2x} $$

x² − 2x − 3 = (x + 1) · h

Now solve for h using synthetic division.

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

$$ \begin{array}{rrrr} \underline{−1}| & 1 & -2 & -3 \\ & & -1 & 3 \\ \hline & 1 & -3 & |\underline{\phantom{0}0} \end{array} $$

The quotient is x − 3 and the remainder is 0. The height of the solid is x − 3.

Practice Exercises

1. What is true if division results in a remainder of zero?

Use long division to divide. State whether the divisor is a factor of the dividend.

2. (x² + 2x − 5) ÷ (x + 2)
3. (x³ + 2x² − 19x + 12) ÷ (x − 3)
4. (3x³ + 2x² − 10x + 19) ÷ (x + 4)
5. (6x³ + 11x² − 13x − 24) ÷ (2x + 3)
6. (3x⁴ − 5x³ + 5x² + 12x − 16) ÷ (x² − 2x + 4)

Use synthetic division to divide. State whether the divisor is a factor of the dividend.

7. (2x³ − 5x² − 13x + 4) ÷ (x − 4)
8. (3x³ + 15x² − 5x − 25) ÷ (x + 5)
9. (5x³ + 2x − 10) ÷ (x − 2)
10. (4x⁴ + 3x³ − 7x² + 15x) ÷ (x + 3)
11. (x⁴ + 2x − 5) ÷ (x − 4)

Use synthetic division to divide with a complex number.

12. (x² + 4) ÷ (x − 2i)
13. (x⁴ − 4x³ + 5x² − 4x + 4) ÷ (x + i)

Problem Solving

14. The volume of a box is 2x³ + 7x² − 4x, the width is x, and the length is x + 4. What is the height of the box?
15. The area of an ellipse is π(2x³ + 5x² + 3x + 2), and the minor axis is x + 2. If the area of an ellipse can be found with A = πab where a is the major axis and b is the minor axis, what is an expression for the length of the major axis?

Mixed Review

16. (2-03) Find the zeros and give the multiplicity of each zero: f(x) = x³ − x² − 9x + 9
17. (2-03) Graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior: f(x) = x³ − x² − 9x + 9
18. (2-02) Find the dimensions of the rectangular corral producing the greatest enclosed area given 150 feet of fencing and using the barn for one side of the corral.
19. (2-01) Multiply (3 − 2i)(4 + 5i)
20. (1-07) Sketch a graph of the function as a transformation of the graph of one of the parent functions: f(x) = −(x + 2)² + 3

Answers

1. The divisor and quotient are factors of the dividend.
2. \(x - \frac{5}{x + 2}\); No
3. x² + 5x − 4; Yes
4. \(3x^2 - 10x + 30 + \frac{-101}{x + 4}\); No
5. 3x² + x − 8; Yes
6. \(3x^2 + x - 5 + \frac{-2x+4}{x^2 - 2x + 4}\); No
7. 2x² + 3x − 1; Yes
8. 3x² − 5; Yes
9. \(5x^2 + 10x + 22 + \frac{34}{x - 2}\); No
10. \(4x^3 - 9x^2 + 20x - 45 + \frac{135}{x + 3}\); No
11. \(x^3 + 4x^2 + 16x + 66 + \frac{259}{x - 4}\); No
12. x + 2i
13. x³ + (−4 − i)x² + (4 + 4i)x − 4i
14. 2x − 1
15. 2x² + x + 1
16. x = − 3, 1, 3
17. ; x-intercepts: (−3, 0), (1, 0), (3, 0) with multiplicity 1; y-intercept: (0, 9); End behavior fall to left and rises to right.
18. 37.5 ft × 75 ft
19. 22 + 7i
20.