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Section6.2Adding and Subtracting Polynomials

Polynomials are a mathematical concept used all around us.

  • A company's sales, \(s\) (in millions of dollars), can be modeled by the equation \(s=2.2t+5.8\text{,}\) where \(t\) stands for the number of years since \(2010\text{.}\)

  • The height of an object from the ground, \(h\) (in feet), launched upward from the top of a building can be modeled by the equation \(h=-16t^2+32t+300\text{,}\) where \(t\) represents the amount of time (in seconds) since the launch.

  • The volume of a particular container, \(v\) (in cubic inches), can be calculated by \(v=\frac{4}{3}\pi r^3+10r^3\text{,}\) where \(r\) stands for the radius (in inches) of the spherical portion of the object.

All of the equations above are polynomials. In this section, we will learn some basic vocabulary relating to polynomials and we'll then learn how to add and subtract polynomials.

Figure6.2.1Alternative Video Lesson

Subsection6.2.1Definition of Polynomial

Definition6.2.2

A polynomial is an expression that is one term or the sum of two or more terms. Some examples of polynomials in one variable would be \(x^2-5x+2\text{,}\) \(x^3-1\text{,}\) and \(7x\text{.}\) The expression \(3x^4y^3+7xy^2-12xy\) would be an example of a polynomial in several variables.

A term of a polynomial is the product of a numerical coefficient and one or more variables raised to non-negative integer exponents. You can view the terms as the individual components that are grouped by multiplication. For example, the polynomial \(x^2-5x+3\) has three terms: \(x^2\text{,}\) \(-5x\) and \(3\text{.}\) The polynomial \(3x^4+7xy^2-12xy\) also has three terms, while \(x^3-1\) has two terms.

The coefficient, or numerical coefficient, of a term is the numerical factor in the term. For example, the coefficient of the term \(\frac{4}{3}x^6\) is \(\frac{4}{3}\text{,}\) the coefficient of the second term of the polynomial \(x^2-5x+3\) is \(-5\text{,}\) and the coefficient of the term \(\frac{x^7}{4}\) is \(\frac{1}{4}\text{.}\)

Example6.2.3

Variables in polynomials must have non-negative integer exponents and a polynomial will never have a variable in the denominator of a fraction or under a square root (or any other radical). Identify which of the following are polynomials and which are not.

  1. The expression \(-2x^9-\frac{7}{13}x^3-1\text{.}\)
  2. The expression \(5x^{-2}-5x^2+3\text{.}\)
  3. The expression \(\sqrt{2}x-\frac{3}{5}\text{.}\)
  4. The expression \(5x^3-5^{-5}x-x^4\text{.}\)
  5. The expression \(\frac{25}{x^2}+23-x\text{.}\)
  6. The expression \(37x^6-x+8^{\frac{4}{3}}\text{.}\)
  7. The expression \(\sqrt{7x}-4x^3\text{.}\)
  8. The expression \(6x^{\frac{3}{2}}+1\text{.}\)
  9. The expression \(6^x-3x^6\text{.}\)
Solution

  1. The expression \(-2x^9-\frac{7}{13}x^3-1\) is a polynomial.
  2. The expression \(5x^{-2}-5x^2+3\) is not a polynomial. Negative exponents on variables make the expression not a polynomial.
  3. The expression \(\sqrt{2}x-\frac{3}{5}\) is a polynomial. Note that the square root is only applied to the 2.
  4. The expression \(5x^3-5^{-5}x-x^4\) is a polynomial. Note that numbers can have negative exponents in polynomials.
  5. The expression \(\frac{25}{x^2}+23-x\) is not a polynomial. Variables are not allowed to be in the denominator in polynomials.
  6. The expression \(37x^6-x+8^{\frac{4}{3}}\) is a polynomial. Note that numbers can have fractional exponents in polynomials.
  7. The expression \(\sqrt{7x}-4x^3\) is not a polynomial. Variables aren't allowed under any radical.
  8. The expression \(6x^{\frac{3}{2}}+1\) is not a polynomial. Exponents must be positive integers, which \(\frac{3}{2}\) is not.
  9. The expression \(6^x-3x^6\) is not a polynomial. Only positive integer exponents are allowed in polynomials, and \(x\) is not an integer since it can change.
Definition6.2.4

The terms of a polynomial are described by their degree. The degree of a term is the number of variable factors in the term. For example, \(5x^2\) is a second degree term, because the exponent \(2\) indicates that there are two \(x\)'s being multiplied in this term. \(-\frac{4}{7}x^5\) is a fifth-degree term, because the exponent \(5\) indicates that there are five \(x\)'s being multiplied in this term. When working with a polynomial in one variable, the exponent on the variable is the degree of the term.

The term with the greatest degree is called the polynomial's leading term and that term's degree is becomes the degree for the polynomial. For example, the degree of the polynomial \(x^2-5x+3\) is two because the leading term is a second-degree term. We can also say that \(x^2-5x+3\) is a second-degree polynomial.

There are some special names for polynomials with certain degrees:

  • A zero-degree polynomial is called a constant polynomial or simply a constant. \(7\) can be viewed as \(7x^0\) and no matter the value of x, the value of the term is constantly \(7\text{.}\)

  • A first-degree polynomial is called a linear polynomial. \(-2x+7\) is an example of a linear polynomial, because \(-2x+7=-2x^1+7\) .

  • A second-degree polynomial is called a quadratic polynomial. \(4x^2-2x+7\) is one such example.

  • A third-degree polynomial is called a cubic polynomial. \(x^3+4x^2-2x+7\) is an example of a cubic polynomial.

Fourth and fifth-degree polynomials are called quartic and quintic polynomials, respectively. If the degree of the polynomial, \(n\text{,}\) is greater than five, we'll simply call it an \(n\)th-degree polynomial.

To help us recognize a polynomial's degree, it is the standard convention to write a polynomial's terms in order from greatest degree term to lowest degree term. When a polynomial is written in this order, it is written in standard form. For example, it is standard practice to write \(7-4x+x^2\) as \(x^2-4x+7\text{,}\) since \(x^2\) is the leading term. By writing the polynomial in standard form, we can look at the first term to determine both the polynomial's degree and leading term.

The degree of a term in a polynomial with more than one variable is still the number of variable factors in the term. \(2x^2y\) is a third-degree term because there are three variable factors in this term: two \(x\)'s and one \(y\text{.}\) The degree of the polynomial is still the degree of the greatest degree term in the polynomial. \(2x^2+3x+2x^2y-10y\) is a third-degree polynomial, because it's leading term's degree is three.

The coefficient of a polynomial's leading term is called the polynomial's leading coefficient. For example, the leading coefficient of \(x^2-5x+3\) is \(1\) (because \(x^2=1\cdot x^2\)).

A term with no variable factor is called a constant term. For example, the constant term of \(x^2-5x+3\) is \(3\text{.}\)

There are special names for polynomials with a small number of terms:

  • A polynomial with one term is called a monomial, such as \(3x^5\) or \(9\text{.}\)

  • A polynomial with two terms is called a binomial, such as \(3x^5+2x\) or \(-2x+1\text{.}\)

  • A polynomial with three terms is called a trinomial, such as \(x^2-5x+3\text{.}\)

There are no special names for polynomials with more than 3 terms.

Exercise6.2.5Definition of Polynomials
Exercise6.2.6Definition of Polynomials

Subsection6.2.2Adding and Subtracting Polynomials

Example6.2.7Production Costs

A particular company only sells one product: local organic jam. The company's production costs only involve two components: supplies and labor. The cost of supplies, \(S\) (in thousands of dollars), can be modeled by \(S=0.05x^2+2x+30\text{,}\) where \(x\) is number of thousands of jars of jam produced. The labor costs, \(L\) (in thousands of dollars), can be modeled by \(0.05x^2+4x\text{,}\) where \(x\) again represents the number of jars produced (in thousands of jars). Find a model for the company's total production costs.

Since this company only has these two costs, we can find a model for the company's total production costs, \(C\) (in thousands of dollars), by adding the supply costs and the labor costs:

\begin{align*} C \amp= \left(0.05x^2+2x+30\right)+\left(0.05x^2+4x\right) \end{align*}

To be able to simplify this expression, we need to combine like terms. In our earlier work, we identified like terms as being terms that have the same variable. In this context, that definition does not work, as four of the terms above have \(x\)'s, but not all four are like terms.

In general, like terms must have the same variable(s) and the variable(s) must have the same exponents. For example, \(16x^3\) and \(4x^2\) are not like terms, while \(-5x^3\) and \(6x^3\) are. \(3xy^2\) and \(8x^2y\) are not like terms, while \(3xy^2\) and \(8xy^2\) are.

To finish simplifying our total revenue model from above, we'll combine the like terms:

\begin{align*} C \amp= \left(0.05x^2+2x+30\right)+\left(0.05x^2+4x\right)\\ \amp= \left(0.05x^2+0.05x^2\right)+\left(2x+4x\right)+30\\ \amp= 0.1x^2+6x+30 \end{align*}

This simplified model can now calculate the total production costs \(C\) (in thousands of dollars) when the company produces \(x\) thousand jars of jam.

Addition of polynomials is really an exercise in recognizing and then combining the like terms.

Example6.2.8

Add \(\left(5x^3+4x^2-6x\right)\) and \(\left(-3x^2+9x-2\right)\text{.}\)

Solution

First identify the like terms and then combine them.

\begin{align*} \left(5x^3+4x^2-6x\right)+\left(-3x^2+9x-2\right) \amp= 5x^3+\left(4x^2-3x^2\right)+\left(-6x+9x\right)-2\\ \amp= 5x^3+x^2+3x-2 \end{align*}
Example6.2.9

Add \(\left(\frac{1}{2}x^2-\frac{2}{3}x-\frac{3}{4}\right)+\left(\frac{3}{2}x^2+\frac{7}{3}x-\frac{1}{4}\right)\text{.}\)

Solution

\begin{align*} \amp\left(\frac{1}{2}x^2-\frac{2}{3}x-\frac{3}{4}\right)+\left(\frac{3}{2}x^2+\frac{7}{3}x-\frac{1}{4}\right)\\ \amp= \left(\frac{1}{2}x^2+\frac{3}{2}x^2\right)+\left(-\frac{2}{3}x+\frac{7}{3}x\right)+\left(-\frac{3}{4}-\frac{1}{4}\right)\\ \amp=\frac{4}{2}x^2 +\frac{5}{3}x-\frac{4}{4}\\ \amp=2x^2 +\frac{5}{3}x-1 \end{align*}
Exercise6.2.10Addition of Polynomials
Example6.2.11Profit, Revenue, and Costs

From our work above, we know the jam company's production costs, \(C\) (in thousands of dollars), for producing \(x\) thousand jars of jam is modeled by \(C=0.1x^2+6x+30\text{.}\) The revenue, \(R\) (in thousands of dollars), from selling the jam can be modeled by \(R=13x\text{,}\) where \(x\) stands for the number of thousands of jars of jam sold. The company's net profit can be calculated using the concept:

\begin{gather*} \text{net profit} = \text{revenue} - \text{costs} \end{gather*}

Assuming all products produced will be sold, find a polynomial to model the company's net profit, \(P\) (in thousands of dollars):

\begin{align*} P \amp= R-C\\ \amp= \left(13x\right)-\left(0.1x^2+6x+30\right)\\ \amp= 13x-0.1x^2-6x-30\\ \amp= -0.1x^2+\left(13x-6x\right)-30\\ \amp=-0.1x^2+7x-30 \end{align*}

Notice that our first step in simplifying this expression was to distribute the subtraction, or \(-1\text{,}\) into the second polynomial \(\left(0.1x^2+6x+30\right)\text{.}\) Once that was done, we could finish simplifying by combining our like terms.

Our last example was an example of polynomial subtraction. As with polynomial addition, polynomial subtraction will be identifying and combining like terms. The difference between the addition and subtraction is that subtraction will first require you to distribute the subtraction (the \(-1\) in front of the parentheses) through the second polynomial. If you are subtracting a polynomial, you must subtract each term of the polynomial.

Example6.2.12

Subtract \(\left(5x^3+4x^2-6x\right)-\left(-3x^2+9x-2\right)\text{.}\)

Solution

We must first distribute the \(-1\) through \(\left(-3x^2+9x-2\right)\) and then we can combine like terms.

\begin{align*} \left(5x^3+4x^2-6x\right)-\left(-3x^2+9x-2\right) \amp= 5x^3+4x^2-6x \highlight{{}+{}} 3x^2 \highlight{{}-{}} 9x \highlight{{}+{}} 2\\ \amp= 5x^3+\left(4x^2+3x^2\right)+\left(-6x-9x\right)+2 \\ \amp= 5x^3+7x^2-15x+2 \end{align*}
Exercise6.2.13Subtraction of Polynomials
Exercise6.2.14Subtraction of Polynomials

Let's look at one more example involving multiple variables. Remember that like terms must have the same variable(s) with the same exponent

Example6.2.15

Subtract \(\left( 3x^2y+8xy^2-17y^3 \right)-\left( -2x^2y+11xy^2+4y^2 \right)\text{.}\)

Solution

Again, we'll begin by distributing the \(-1\) through \(\left( -2x^2y+11xy^2+4y^2 \right)\text{.}\) Once we've done this, we'll need to identify and combine like terms.

\begin{align*} \amp\left( 3x^2y+8xy^2-17y^3 \right)-\left( -2x^2y+11xy^2+4y^2 \right)\\ \amp= 3x^2y+8xy^2-17y^3 \highlight{{}+{}} 2x^2y \highlight{{}-{}} 11xy^2 \highlight{{}-{}} 4y^2\\ \amp=\left(3x^2y+2x^2y\right)+\left(8xy^2-11xy^2\right)-17y^3 -4y^2\\ \amp= 5x^2y-3xy^2-17y^3-4y^2 \end{align*}

\(3x^2y\) and \(2x^2y\) are like terms, as they both have \(x^2y\text{.}\)

\(8xy^2\) and \(-11xy^2\) are like terms, as they both have \(xy^2\text{.}\)

\(-17y^3\) and \(-4y^2\) are not like terms, as their exponents are not the same.

Subsection6.2.3Exercises

Simplifying Polynomials

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WeBWorK Exercise
2
WeBWorK Exercise
3
WeBWorK Exercise
4
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5
WeBWorK Exercise
6
WeBWorK Exercise
7
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WeBWorK Exercise

Applications of Simplifying Polynomials

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WeBWorK Exercise
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