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Elementary Algebra

Katherine Yoshiwara

Section 7.1 Polynomials

Subsection What is a Polynomial?

In previous Lessons we have studied linear (first-degree) expressions, such as the formula for tuition if you take \(u\) units,
\begin{equation*} T= 50 + 15u \end{equation*}
and quadratic (second-degree) expressions, such as the height of an algebra book thrown off a cliff,
\begin{equation*} h=-16t^2 + 32t + 240 \end{equation*}
We have even encountered some cubic or third-degree expressions, such as the formula for the volume of a sphere,
\begin{equation*} V=\dfrac{4}{3}\pi r^3 \end{equation*}
All of these expressions are part of a larger class of mathematical objects called polynomials. In this section we introduce some terminology used to discuss polynomials and review the algebraic skills used to study them.

Polynomial.

A polynomial is an algebraic expression with several terms. Each term is a power of a variable (or a product of powers) with a constant coefficient.

Example 7.1.

The following expressions are polynomials.
\(4x^3+2x^2-7x+5\) \(\hphantom{0000}\) \(\dfrac{1}{2}at^2+vt\)
\(2a^2-6ab+3b^2\) \(\hphantom{0000}\) \(\pi r^2h\)

Look Closer.

The exponents in a polynomial must be whole numbers, which means that a polynomial has no radicals containing variables, and no variables in the denominators of fractions.

Number of Terms.

Recall that: An algebraic expression with only one term is called a monomial. An expression with two terms is called a binomial, and an expression with three terms is a trinomial.

Example 7.2.

  • The expression \(2x^3\) is a monomial (it has only one term).
  • The expression \(x^2-16\) is a binomial (it has two terms).
  • The expression \(ax^2+bx+c\) is called a quadratic trinomial.

QuickCheck 7.3.

What is a polynomial?
Answer.
An algebraic expression with several terms. Each term is a power of a variable (or a product of powers) with a constant coefficient.

Subsection Degree

In a term containing only one variable, the exponent of the variable is called the degree of the term. (The degree of a constant term is zero, for reasons that will become clear in Chapter 9.) For example,
\begin{align*} \amp 3x^2~~~~~~ \amp \amp\text{is of second degree}\\ \amp 8y^3~~~~~~ \amp \amp\text{is of third degree}\\ \amp 4x~~~~~~ \amp \amp\text{is of first degree (the exponent on}~x~ \text{ is 1)}\\ \amp 5 ~~~~~~ \amp \amp\text{is of zero degree} \end{align*}
The degree of a polynomial in one variable is the largest exponent that appears in any term.

Example 7.4.

The degree of a polynomial does not depend on the number of terms in the polynomial.
\begin{align*} \amp 2x+1~~~~~~ \amp \amp\text{is of first degree}\\ \amp 3y^2-2y+2~~~~~~ \amp \amp\text{is of second degree}\\ \amp m-2m^2+m^5~~~~~~ \amp \amp\text{is of fifth degree} \end{align*}

QuickCheck 7.5.

What is the difference between a polynomial of degree four and a polynomial with four terms?
Answer.
Degree four means the largest exponent is four.
Polynomials in one variable are usually written in descending powers of the variable. The term with the largest exponent comes first, then the term with the next highest exponent, and so on down to the constant term, if there is one.

Example 7.6.

Write the polynomial \(~x+3x^4-5-2x^2~\) in descending powers.

Solution.

We start with the highest power. We must be careful to keep the correct sign of each term.
graphic

QuickCheck 7.7.

What does it mean to write a polynomial in descending powers?
Answer.
Start with the term with the largest exponent, and continue with decreasing exponents.

Subsection Evaluating Polynomials

We evaluate polynomials the same way we evaluate any other algebraic expression: by substituting the given values for the variables and then following the order of operations to simplify.

Example 7.8.

Evaluate \(~16t^3-6t+20~\) for \(~t=\dfrac{3}{2}\text{.}\)

Solution.

Substitute \(\alert{\dfrac{3}{2}}\) for \(t\text{,}\) and follow the order of operations to simplify.
\begin{align*} 16\left(\alert{\dfrac{3}{2}}\right)^3 \amp -6(\alert{\dfrac{3}{2}})+20 \amp \amp \blert{\text{Compute the power.}}\\ \amp = 16(\dfrac{27}{8})-6\left(\dfrac{3}{2}\right)+20 \amp \amp \blert{\text{Perform all multiplications.}}\\ \amp = 54-9+20 \amp \amp \blert{\text{Add.}}\\ \amp = 65 \end{align*}

Subsection Like Terms

Remember that like terms are any terms that are exactly alike in their variable factors. This means that the exponents on the variable factors must also match. For example,
\begin{align*} x^3~~~\amp \text{and}~~~2x+1 \amp \amp \blert{\text{Not like terms}}\\ 2x^2y~~~\amp \text{and}~~~3y^2-2y+2 \amp \amp \blert{\text{Not like terms}} \end{align*}
are not like terms because their variable factors are different.
However,
\begin{align*} \dfrac{1}{2}x^2y~~~\amp \text{and}~~~-3yx^2 \amp \amp \blert{\text{Like terms}} \end{align*}
are like terms, because \(x^2y\) and \(yx^2\) are equivalent expressions.

Combining Like Terms.

Recall that: To add or subtract like terms, we add or subtract their numerical coefficients. The variable factors of the terms remain unchanged.

QuickCheck 7.9.

What is the numerical coefficient of a term?
Answer.
The factor in a term that is a constant, not a variable.

Look Closer.

Which of the following two expressions can be simplified?
\begin{equation*} 3x^2+5x^3~~~~~~\text{and}~~~~~~3x^2+5x^2 \end{equation*}
In the first expression, the two terms have different exponents, even though the base, \(x\text{,}\) is the same. Powers of the same variable with different exponents are not like terms. The first expression, \(3x^2+5x^3\text{,}\) cannot be simplified.
In the second expression, \(3x^2\) and \(5x^2\) are like terms.Thus,
\begin{equation*} 3x^2+5x^2=8x^2 \end{equation*}
Note that we do not change the exponent on \(x\) in the sum; it is still 2.

Example 7.10.

Simplify by combining any like terms:
\begin{equation*} x^3+2x^2-2x^3-(-4x^2)+4x \end{equation*}

Solution.

The terms \(x^3\) and \(-2x^3\) are like terms; so are \(2x^2\) and \(-4x^2\text{.}\) We group the like terms together and combine them.
\begin{equation*} \blert{x^3-2x^3} + \alert{2x^2-(-4x^2)} + 4x = \blert{-x^3} + \alert{6x^2} + 4x \end{equation*}
The last term, \(4x\text{,}\) is not combined with any other terms.

Subsection Adding Polynomials

To add two polynomials we remove parentheses and combine like terms.

Example 7.11.

\begin{align*} (4a^3 \amp -2a^2-3a+1)+(2a^3+4a-5) \amp \amp \blert{\text{Remove parentheses.}}\\ \amp = \blert{4a^3}-2a^2 \alert{-3a}+1 \blert{+2a^3} \alert{+4a}-5 \amp \amp \blert{\text{Combine like terms.}}\\ \amp = \blert{6a^3}-2a^2 \alert{+a}-4 \end{align*}
If there are several terms in the polynomials, it is often easier to use a vertical format for addition, as shown in the next example.

Example 7.12.

Add \(~4x^2+2-7x~\) and \(~5x-5+2x^2\)

Solution.

We first write each polynomial in descending powers of \(x\text{.}\) Then we write the second polynomial beneath the first, aligning like terms.
\begin{align*} \amp 4x^2-7x+2 \amp \amp \blert{\text{Combine like terms vertically.}}\\ + ~~\amp \underline{2x^2+5x-5} \\ \amp 6x^2-2x-3 \end{align*}

Subsection Subtracting Polynomials

Recall that if an expression in parentheses is preceded by a minus sign, we must change the sign of each term within parentheses when we remove the parentheses. This rule applies when we subtract polynomials.

Example 7.13.

Subtract the polynomials \(~(4x^2+2x-5)-(2x^2-3x-2)\)

Solution.

We first remove the parentheses, remembering to change the sign of each term of the subtracted polynomial.
\begin{align*} (4x^2 \amp +2x-5)-(2x^2-3x-2) \amp \amp \blert{\text{Change signs.}}\\ \amp = 4x^2+2x-5 \blert{-} 2x^2\blert{+}3x\blert{+}2\amp \amp \blert{\text{Combine like terms.}}\\ \amp = 2x^2+5x-3 \end{align*}

Caution 7.14.

In Example 7.13, the following second step is \(\alert{\text{incorrect}}\text{:}\)
\begin{equation*} 4x^2+2x-5-2x^2-3x-2 \end{equation*}
because we have not changed the sign of \(\alert{\text{each}}\) term inside parentheses.
We can also use a vertical format for subtraction. In this technique, we change the sign of each term in the bottom, or subtracted, polynomial, as shown below.

Example 7.15.

Subtract \(~2n^2-3n-2~\) from \(~4n^2+2n-5\)

Solution.

\(~~~~~~~~4n^2+2n-5\) \(\hphantom{0000}\) \(~~~~~~~~~~4n^2+2n-5\)
\(-~~\underline{(2n^2-3n-2)}\) \(\blert{\rightarrow~~\text{Change signs of each term.}~~\rightarrow}\) \(+~~~\underline{-2n^2+3n+2}\)
\(\hphantom{0000}\) \(\hphantom{0000}\) \(~~~~~~~~~~2n^2+5n-3\)

QuickCheck 7.16.

When subtracting two polynomials, what must we do before combining like terms?
Answer.
Change the sign of each term in the subtracted polynomial.

Subsection Application

Applied problems may involve addition or subtraction of polynomials. In particular, to calculate the profit it earns by selling its product, a company must subtract the cost of producing the goods from the revenue it earns by selling them. We can state this as a formula.

Profit.

\begin{equation*} \blert{\text{Profit = Revenue - Cost}} \end{equation*}

Example 7.17.

It costs The Cookie Company \(~200+2x~\) dollars to produce \(x\) bags of cookies per week, and they earn \(~8x-0.01x^2~\) dollars from the sale of \(x\) bags of cookies.
  1. Write a polynomial for the profit earned by The Cookie Company on \(x\) bags of cookies.
  2. Find the company’s profit on 300 bags of cookies, and on 600 bags.

Solution.

  1. Applying the profit formula to this situation, we subtract polynomials to find
    \begin{align*} \text{Profit} \amp =(8x-0.01x^2)-(200+2x) \amp \amp \blert{\text{Change signs.}}\\ \amp = 8x-0.01x^2-200-2x\amp \amp \blert{\text{Combine like terms.}}\\ \amp = -0.01x^2+6x-200 \end{align*}
  2. Evaluate the profit polynomial for \(x=\alert{300}\) and for \(x=\blert{600}\text{.}\) For 300 bags of cookies, the profit is
    \begin{equation*} -0.01(\alert{300})^2+6(\alert{300})-200 = -900+1800-200=700 \end{equation*}
    or $700. For 600 bags of cookies, the profit is
    \begin{equation*} -0.01(\blert{600})^2+6(\blert{600})-200 = -3600+3600-200=-200 \end{equation*}
    The company loses $200 if they produce 600 bags of cookies.

Skills Warm-Up 7.18.

Replace the comma in each pair by the proper symbol: \(~\gt,~ \lt,~\) or \(~=\text{.}\)
  1. \(\displaystyle (-2)^3,~-2^3\)
  2. \(\displaystyle -5^4,~(-5)^4\)
  3. \(\displaystyle 7-3^2,~4^2\)
  4. \(\displaystyle 2+4^3,~6^3\)
  5. \(\displaystyle 4 \cdot 2^3,~8^3\)
  6. \(\displaystyle 2 \cdot 5^2,~10^2\)
  7. \(\displaystyle \dfrac{1}{2},~\left(\dfrac{1}{2}\right)^2\)
  8. \(\displaystyle \dfrac{3}{4},~\left(\dfrac{3}{4}\right)^2\)

Answer.

  1. \(\displaystyle =\)
  2. \(\displaystyle \lt\)
  3. \(\displaystyle \lt\)
  4. \(\displaystyle \lt\)
  5. \(\displaystyle \lt\)
  6. \(\displaystyle \lt\)
  7. \(\displaystyle \gt\)
  8. \(\displaystyle \gt\)

Subsection Lesson

Activity 7.1. Polynomials.

  1. Which of the following expressions are polynomials?
    1. \(\displaystyle t^2-7t+3\)
    2. \(\displaystyle 2+\dfrac{6}{x}\)
    3. \(\displaystyle -a^3b^2c^4\)
    4. \(\displaystyle 3w^2-w+5\sqrt{w}\)
  2. Give the degree of each polynomial.
    1. \(\displaystyle 5x^9-1\)
    2. \(\displaystyle 3t^2-t-6t^4+8\)
  3. Write the polynomial \(~3x-8-6x^3-x^6~\) in descending powers of \(x\text{.}\)
  4. Evaluate \(~2x^3y^2-3xy^3-3~\) for \(x=-2,~y=-3\text{.}\)

Activity 7.2. Adding and Subtracting Polynomials.

  1. Simplify by combining any like terms.
    \begin{equation*} 2a^2b-5ba+4ba^29(-3ab) \end{equation*}
  2. Add the polynomials.
    \begin{equation*} (2m^4-4m^3+3m-1)+(m^3+4m^2-6m+2) \end{equation*}
  3. Use a vertical format to add \(~5y^2-3y+1~\) to \(~-3y^2+6y-7\)
  4. Subtract the polynomials.
    \begin{equation*} (2m^4-4m^3+3m-1)-(m^3+4m^2-6m+2) \end{equation*}
  5. Use a vertical format to subtract \(~2x^2+5-2x~\) from \(~7-3x-4x^2\)
  6. Simplify \(~2a^2(3-a+4a^2)-3a(5a-a^2)\)

Activity 7.3. Applications.

  1. The length of a box is twice its width, \(w\text{,}\) and its height is 3 inches less than its width.
    1. Sketch a picture of the box, and label its sides.
    2. Write a polynomial for the volume of the box.
    3. Write a polynomial for the surface area of the box.
  2. Milly’s Muffins can sell \(x\) muffins at the farmers’ market if she charges \(~6-0.02x~\) dollars for a muffin. It costs Milly \(~50+1.5x~\) dollars to make \(x\) muffins.
    1. Write a polynomial for Milly’s revenue from selling muffins.
      Hint: Revenue \(~=~\)(number of items sold)\(~\times ~\)(price per item)
    2. Write a polynomial for Milly’s profit.
    3. Find Milly’s profit if she sells 50 muffins, 100 muffins, and 150 muffins.

Subsubsection Wrap-Up

Objectives.
In this Lesson we practiced the following skills:
  • Identifying a polynomial
  • Identifying degree and number of terms
  • Writing a polynomial in descending powers
  • Evaluating a polynomial
  • Adding and subtracting polynomials
  • Writing simple polynomial expressions
Questions.
  1. Name two places that a variable cannot appear in a polynomial.
  2. What do you suppose "ascending powers" means?
  3. What is wrong with this statement:
    \begin{equation*} 5x-(2x+3x^2) = 5x=2x+3x^2 \end{equation*}

Activity 7.4. HOmework Preview.

  1. Explain why each expression is not a polynomial.
    1. \(\displaystyle 4x^3-x^2+\dfrac{3}{x}\)
    2. \(\displaystyle 2x^2-3x\sqrt{x} + 5x\)
  2. Evaluate \(~2x^3-4x^2+3x-6~\) for \(x=-3\text{.}\)
  3. Evaluate \(~\dfrac{5}{4}x^2+\dfrac{1}{2}xy-2y^2~\) for \(x=-4,~y=2\text{.}\)
  4. Add \(~(5x^3-3x+4x^2-1)+(3+2x-x^3-6x^2)\)
  5. Subtract \(~(t^3-2t^2-3t+5)-(2t^3+4t-3-t^2)\)
  6. Simplify \(~ab(3ab-b)-b^2(2a^2-ab)\)
Answers to Homework Preview
    1. It has a variable in the denominator
    2. It has a variable under a radical
  2. \(\displaystyle -105\)
  3. \(\displaystyle 8\)
  4. \(\displaystyle 4x^3-2x^2-x+2\)
  5. \(\displaystyle -t^3-t^2-7t+8\)
  6. \(\displaystyle a^2b^2-ab^2+ab^3\)

Exercises Homework 7.1

1.

Which of the following are not allowed in a polynomial?
  1. More than three terms
  2. Coefficients that are fractions
  3. Division by a variable
  4. A term without variables

2.

Give an example of each type of polynomial.
  1. A monomial of fourth degree
  2. A binomial of first degree
  3. A trinomial of degree 2
  4. A monomial of degree 0

Exercise Group.

Which of the expressions in Problems 3–4 are polynomials? If it is not a polynomial, explain why not.
3.
  1. \(\displaystyle 5x^4-3x^2\)
  2. \(\displaystyle 3x+1+\dfrac{2}{x^2}\)
  3. \(\displaystyle \dfrac{1}{2a^2+5a-6}\)
  4. \(\displaystyle \dfrac{2}{3}t^2+\dfrac{1}{4}t^3+\dfrac{5}{8}\)
4.
  1. \(\displaystyle 27y^8\)
  2. \(\displaystyle 2a^2-6ab+3b^2\)
  3. \(\displaystyle \dfrac{3}{x^4}-\dfrac{7}{x^3}+\dfrac{5}{3}\)
  4. \(\displaystyle 9z^2-\dfrac{1}{2}z^2+8z^6\)

5.

Give the degree of each polynomial.
  1. \(\displaystyle x^2+4x-\dfrac{1}{4}\)
  2. \(\displaystyle y-2.8y^7\)
  3. \(\displaystyle \dfrac{z}{4}-3z^4+4z^3\)

Exercise Group.

For Problems 6–7, write the polynomial in descending powers of \(x\text{.}\)
6.
\(x-1.9x^3+6.4\)
7.
\(6xy-2x^2+2y^3\)

Exercise Group.

For Problems 8–11, evaluate the polynomial for the given values of the variables.
8.
\(2-z^2-2z^3~~\) for \(~z=-2\)
9.
\(2a^4+3a^2-3a~~\) for \(~a=1.6\)
10.
\(-abc^2~~\) for \(~a=-3,~b=2,~c=2\)
11.
\(x^2-3x+2~~\) for \(~x=\sqrt{3}\)

Exercise Group.

For Problems 12–13, simplify by combining like terms.
12.
\(6b^3-2b^3-(-8b^3)\)
13.
\(6x-3y+5xy-6y+xy\)

Exercise Group.

For Problems 14–16, explain why the calculation is incorrect, and give the correct answer.
14.
\(6w^3+8w^3~ \rightarrow ~14w^6\)
15.
\(6+3x^2~ \rightarrow ~9x^2\)
16.
\(4t^2+7-(3t^2-5)~ \rightarrow ~t^2+2\)

Exercise Group.

For Problems 17–18, add or subtract the polynomials.
17.
\((2y^3-4y^2-y)+(6y^2+2y+1)\)
18.
\((5x^3+3x^2-4x+8)-(2x^3-4x-3)\)

Exercise Group.

For Problems 19–20, use a vertical format to add or subtract the polynomials.
19.
Add \(~8x^2-3x+4~\) to \(~-2x^2+5x-7\)
20.
Subtract \(~4x^2-3x-1~\) from \(~-3x^2+4x-2\)

21.

Brenda is flying at an altitude of 4000 feet when she starts her descent for landing. After traveling \(x\) miles horizontally, her altitude is given in feet by
\begin{equation*} h=125x^3-750x^2+4000 \end{equation*}
  1. What is Brenda’s altitude after traveling 2 miles horizontally?
  2. Evaluate the polynomial for \(x=4\text{.}\) What does this mean?

22.

The height of a box is 4 inches less than its width, \(w\text{,}\) and its length is 8 inches greater than its width. Write a polynomial for the volume of the box.

23.

Suppose you want to choose 3 items from a list of \(n\) possible items. The number of different ways you can make your choice is given by the polynomial
\begin{equation*} \dfrac{1}{6}n^3-\dfrac{1}{2}n^2+\dfrac{1}{3}n \end{equation*}
  1. How many ways can you pick 3 elective courses from a list of 8 approved courses?
  2. How many ways can you pick 3 cards from a deck of 52?
  3. How many different 3-person committees can be chosen from a club with 20 members?

24.

After you apply the brakes, a small car traveling at \(s\) miles per hour can stop in approximately \(~0.04s^2+0.6s~\) feet. Will a car traveling at 50 miles per hour on the freeway avoid hitting a stalled car in the same lane 100 feet ahead?

25.

Evaluate each polynomial for \(n=10\text{.}\) Try to do the calculations mentally. What do you notice?
  1. \(\displaystyle 5n^2+6n+7\)
  2. \(\displaystyle 5n^3+n^2+3n+3\)
  3. \(\displaystyle n^3+1\)
  4. \(\displaystyle 8n^4+8n\)

26.

Expand each expression by removing parentheses. What do you notice?
  1. \(\displaystyle x[x(x+3)+4]+1\)
  2. \(\displaystyle x(x[x(x-7)-5]+8)-3\)

27.

  1. Evaluate the expression in Problem 26a for \(x=2\text{.}\) Can you do this mentally? Is it easier to evaluate the expression before or after expanding it?
  2. Use a calculator to evaluate the expression for \(x=0.8\text{.}\)

28.

GreatOutdoors sells specialty tents for high-altitude camping. Their cost for producing \(x\) tents is given by
\begin{equation*} C=x^3-12x^2+80x+180 \end{equation*}
and their revenue from selling \(x\) tents is
\begin{equation*} R=2800x-2x^2 \end{equation*}
  1. Write a polynomial for the profit GreatOutdoors makes from selling \(x\) tents. (Hint: Recall that Profit \(=\)Revenue\(-\)Cost.)
  2. Find GreatOutdoors’ profit from selling 10 tents, 20 tents, and 50 tents.

29.

The Flying Linguine Brothers are working on a new act for the circus. Mario swings from a trapeze and catches Alfredo, who has somersaulted off a trampoline, in mid-air. Mario’s height at time \(t\) seconds is given approximately by \(~12t^2-24t+34~\) feet, and Alfredo’s height at time \(t\) is \(~-16t^2+32t+6~\text{.}\)
  1. Write a polynomial for the difference in height between Mario and Alfredo at any time \(t\text{.}\)
  2. Find the difference in height between Mario and Alfredo when they start (at time \(t=0\)), and after \(\dfrac{1}{2}\) second.
  3. When will Mario and Alfredo be at the same height?

30.

Ralph and Waldo start in towns that are 20 miles apart and travel in opposite directions for 2 hours. Ralph travels 30 miles per hour faster than Waldo. Let \(w\) stand for Waldo’s speed and write algebraic expressions to answer the following questions.
  1. How far did Waldo travel?
  2. What was Ralph’s speed?
  3. How far did Ralph travel?
  4. What is the distance between Ralph and Waldo after the 2 hours?

31.

Let \(T_m\) stand for the sum of the squares of the first \(m\) integers. For example,
\begin{align*} T_1 \amp = 1^2=1\\ T_2 \amp = 1^2+2^2=5\\ T_2 \amp = 1^2+2^3+3^2=14 \end{align*}
and so on.
  1. Fill in the table showing the first 10 values of \(T_m\text{.}\)
  2. Evaluate the polynomial
    \begin{equation*} \dfrac{1}{3}m^3+\dfrac{1}{2}m^2+\dfrac{1}{6}m \end{equation*}
    for integer values of \(m\) from 1 to 10, and fill in the table.
  3. Compare your answers to parts (a) and (b).
\(~m~\) \(~T_m~\) \(\dfrac{1}{3}m^3+\dfrac{1}{2}m^2+\dfrac{1}{6}m\)
\(1\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(2\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(3\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(4\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(5\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(6\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(7\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(8\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(9\) \(\hphantom{0000}\) \(\hphantom{0000}\)
\(10\) \(\hphantom{0000}\) \(\hphantom{0000}\)

32.

Evaluate
\begin{equation*} (2x+y-z)+(3x-4y+6z)-(5x-9y-3z) \end{equation*}
for \(~x=2.8,~y=-3.6,~z=1.8~\)
(Hint: There is a hard way and an easy way.)
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