Section6.2Adding and Subtracting Polynomials
¶A polynomial is a particular type of mathematical expression used for things all around us.
A company's sales, \(s\) (in millions of dollars), can be modeled by \(2.2t+5.8\text{,}\) 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 \(16t^2+32t+300\text{,}\) where \(t\) represents the amount of time (in seconds) since the launch.
The volume of an opentop box with a square base, \(V\) (in cubic inches), can be calculated by \(30s^2\frac{1}{2}s^2\text{,}\) where \(s\) stands for the length of the square base, and the box sides have to be cut from a certain square piece of metal.
All of the expressions 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.
Subsection6.2.1Polynomial Vocabulary
Definition6.2.2
A polynomial is an expression that consists of terms summed together. Each term must be the product of a number and one or more variables raised to whole number powers. Since \(0\) is a whole number, a term can just be a number. A polynomial may have just one term. The expression \(0\) is also considered a polynomial, with zero terms.
Some examples of polynomials in one variable are:
The expression \(3x^4y^3+7xy^212xy\) is an example of a polynomial in several variables.
Definition6.2.3
A term of a polynomial is the product of a numerical coefficient and one or more variables raised to whole number powers. Since \(0\) is a whole number, a term can just be a number.
For example:
the polynomial \(x^25x+3\) has three terms: \(x^2\text{,}\) \(5x\text{,}\) and \(3\text{;}\)
the polynomial \(3x^4+7xy^212xy\) also has three terms;
the polynomial \(t^31\) has two terms.
Definition6.2.4
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 \(x^25x+3\) is \(5\text{;}\)
the coefficient of the term \(\frac{y^7}{4}\) is \(\frac{1}{4}\text{.}\)
Remark6.2.5
Because variables in polynomials must have whole number exponents, a polynomial will never have a variable in the denominator of a fraction or under a square root (or any other radical).
Checkpoint6.2.6
Definition6.2.7
When a term only has one variable, its degree is the exponent on that variable. When a term has more than on variable, its degree is the sum of the exponents on the variables in that term. When a term has no variables, its degree is \(0\text{.}\)
For example:
the degree of \(5x^2\) is \(2\text{;}\)
the degree of \(\frac{4}{7}y^5\) is \(5\text{.}\)
the degree of \(4x^2y^3\) is \(5\text{.}\)
Polynomial terms are often classified by their degree. In doing so, we would refer to \(5x^2\) as a seconddegree term.
Definition6.2.8
The degree of a polynomial is the greatest degree that appears amongst its terms. If the polynomial is just \(0\text{,}\) it has no terms, and we say its degree is \(1\text{.}\)
The leading term of a polynomial is the term with the greatest degree (assuming there is one, and there is no tie).
For example, the degree of the polynomial \(x^25x+3\) is \(2\) because the terms have degrees \(2\text{,}\) \(1\text{,}\) and \(0\text{,}\) and \(2\) is the largest. Its leading term is \(x^2\text{.}\) Polynomials are often classified by their degree, and we would say that \(x^25x+3\) is a seconddegree polynomial.
The coefficient of a polynomial's leading term is called the polynomial's leading coefficient. For example, the leading coefficient of \(x^25x+3\) is \(1\) (because \(x^2=1\cdot x^2\)).
Definition6.2.9
A term with no variable factor is called a constant term. For example, the constant term of \(x^25x+3\) is \(3\text{.}\)
For example, the constant term of the polynomial \(x^25x+3\) is \(3\text{.}\)
There are some special names for polynomials with certain degrees:

A zerodegree polynomial is called a constant polynomial or simply a constant.
An example is the polynomial \(7\text{,}\) which has degree zero because it can be viewed as \(7x^0\text{.}\)

A firstdegree polynomial is called a linear polynomial.
An example is \(2x+7\text{.}\)

A seconddegree polynomial is called a quadratic polynomial.
An example is \(4x^22x+7\text{.}\)

A thirddegree polynomial is called a cubic polynomial.
An example is \(x^3+4x^22x+7\text{.}\)
Fourthdegree and fifthdegree 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 \(n\)thdegree polynomial. For example, the polynomial \(5x^84x^5+1\) is an \(8\)thdegree polynomial.
Remark6.2.10
To help us recognize a polynomial's degree, it is the standard convention to write a polynomial's terms in order from greatestdegree term to lowestdegree term. When a polynomial is written in this order, it is written in standard form. For example, it is standard practice to write \(74xx^2\) as \(x^24x+7\) 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.
There are special names for polynomials with a small number of terms:
A polynomial with one term, such as \(3x^5\) or \(9\text{,}\) is called a monomial.
A polynomial with two terms, such as \(3x^5+2x\) or \(2x+1\text{,}\) is called a binomial.
A polynomial with three terms, such as \(x^25x+3\text{,}\) is called a trinomial.
Subsection6.2.2Adding and Subtracting Polynomials
Example6.2.11Production Costs
A particular company only sells one product: ketchup. 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 ketchup produced. The labor costs, \(L\) (in thousands of dollars), can be modeled by \(0.1x^2+4x\text{,}\) 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:
To finish simplifying our total revenue model from above, we'll combine the like terms:
This simplified model can now calculate the total production costs \(C\) (in thousands of dollars) when the company produces \(x\) thousand jars of ketchup.
In short, the process of adding two or more polynomials involves recognizing and then combining the like terms.
Checkpoint6.2.12
Example6.2.13
Add \(\left(\frac{1}{2}x^2\frac{2}{3}x\frac{3}{2}\right)+\left(\frac{3}{2}x^2+\frac{7}{2}x\frac{1}{4}\right)\text{.}\)
Example6.2.14Profit, Revenue, and Costs
From Example 6.2.11, we know the ketchup company's production costs, \(C\) (in thousands of dollars), for producing \(x\) thousand jars of ketchup is modeled by \(C=0.15x^2+6x+30\text{.}\) The revenue, \(R\) (in thousands of dollars), from selling the ketchup can be modeled by \(R=13x\text{,}\) where \(x\) stands for the number of thousands of jars of ketchup sold. The company's net profit can be calculated using the concept:
Assuming all products produced will be sold, a polynomial to model the company's net profit, \(P\) (in thousands of dollars) is:
The key distinction between the addition and subtraction of polynomials is that when we subtract a polynomial, we must subtract each term in that polynomial.
Remark6.2.15
Notice that our first step in simplifying the expression in Example 6.2.14 was to subtract every term in the second expression. We can also think of this as distributing a factor of \(1\) across the second polynomial, \(0.15x^2+6x+30\text{,}\) and then adding these terms as follows:
Example6.2.16
Subtract \(\left(5x^3+4x^26x\right)\left(3x^2+9x2\right)\text{.}\)
We must first subtract every term in \(\left(3x^2+9x2\right)\) from \(\left(5x^3+4x^26x\right)\text{.}\) Then we can combine like terms.
Checkpoint6.2.17
Let's look at one more example involving multiple variables. Remember that like terms must have the same variable(s) with the same exponent.
Example6.2.18
Subtract \(\left( 3x^2y+8xy^217y^3 \right)\left(2x^2y+11xy^2+4y^2 \right)\text{.}\)
Again, we'll begin by subtracting each term in \(\left(2x^2y+11xy^2+4y^2\right)\text{.}\) Once we've done this, we'll need to identify and combine like terms.
SubsectionExercises
Vocabulary Questions
1
Is the following expression a monomial, binomial, or trinomial?
\({8t^{2}+12t}\)
monomial
binomial
trinomial
What is the degree of the expression?
2
Is the following expression a monomial, binomial, or trinomial?
\({18t^{16}12t^{13}}\)
monomial
binomial
trinomial
What is the degree of the expression?
3
Is the following expression a monomial, binomial, or trinomial?
\({10}\)
monomial
binomial
trinomial
What is the degree of the expression?
4
Is the following expression a monomial, binomial, or trinomial?
\({25}\)
monomial
binomial
trinomial
What is the degree of the expression?
5
Is the following expression a monomial, binomial, or trinomial?
\({17y^{6}+19y^{5}2y^{4}}\)
monomial
binomial
trinomial
What is the degree of the expression?
6
Is the following expression a monomial, binomial, or trinomial?
\({2y^{6}+8y^{4}+15y^{2}}\)
monomial
binomial
trinomial
What is the degree of the expression?
7
Is the following expression a monomial, binomial, or trinomial?
\({12y^{3}9y^{5}4y}\)
monomial
binomial
trinomial
What is the degree of the expression?
8
Is the following expression a monomial, binomial, or trinomial?
\({15r^{4}+8r^{5}15r^{2}}\)
monomial
binomial
trinomial
What is the degree of the expression?
9
Is the following expression a monomial, binomial, or trinomial?
\({r^{3}}\)
monomial
binomial
trinomial
What is the degree of the expression?
10
Is the following expression a monomial, binomial, or trinomial?
\({14t^{11}}\)
monomial
binomial
trinomial
What is the degree of the expression?
11
Find the degree of the following polynomial
\(\displaystyle{ {18x^{5}y^{5}+13x^{4}y^{2}+19x^{2}8} }\)
The degree of this polynomial is .
12
Find the degree of the following polynomial
\(\displaystyle{ {19x^{5}y^{7}x^{4}y^{4}5x^{2}19} }\)
The degree of this polynomial is .
Simplifying Polynomials
13
Add the two binomials.
\(\left({7x8}\right)+\left({6x+5}\right)\)
14
Add the two binomials.
\(\left({5x+6}\right)+\left({6x1}\right)\)
15
Add the two binomials.
\(\left({3x^{2}2x}\right)+\left({3x^{2}6x}\right)\)
16
Add the two binomials.
\(\left({9x^{2}10x}\right)+\left({9x^{2}10x}\right)\)
17
Add the two trinomials.
\(\left({7x^{2}+6x+10}\right)+\left({10x^{2}9x+9}\right)\)
18
Add the two trinomials.
\(\left({8x^{2}9x+10}\right)+\left({6x^{2}4x3}\right)\)
19
Add the two trinomials.
\(\left({2t^{3}+2t^{2}+10}\right)+\left({6t^{3}+7t^{2}+2}\right)\)
20
Add the two trinomials.
\(\left({8t^{3}+9t^{2}1}\right)+\left({7t^{3}10t^{2}5}\right)\)
21
Add the two trinomials.
\(\left({5x^{6}+6x^{4}+10x^{2}}\right)+\left({8x^{6}+4x^{4}+8x^{2}}\right)\)
22
Add the two trinomials.
\(\left({10x^{6}+4x^{4}x^{2}}\right)+\left({9x^{6}6x^{4}8x^{2}}\right)\)
23
Add the two polynomials.
\(\left({0.3y^{5}+0.9y^{4}+0.8y^{2}+0.9}\right)+\left({0.7y^{5}0.5y^{3}+0.7}\right)\)
24
Add the two polynomials.
\(\left({0.4y^{5}0.2y^{4}+0.4y^{2}+0.8}\right)+\left({0.1y^{5}+0.9y^{3}+0.3}\right)\)
25
Add the two polynomials
\(\left({6x^{3}9x^{2}+9x+{\frac{1}{2}}}\right)+\left({6x^{3}5x^{2}3x+{\frac{5}{4}}}\right)\)
26
Add the two polynomials
\(\left({7x^{3}3x^{2}9x+{\frac{1}{6}}}\right)+\left({8x^{3}2x^{2}7x+{\frac{1}{2}}}\right)\)
27
Subtract the two binomials.
\(\left({4x5}\right)\left({3x+8}\right)\)
28
Subtract the two binomials.
\(\left({7x+9}\right)\left({10x+2}\right)\)
29
Subtract the two binomials.
\(\left({9x^{2}+x}\right)\left({x^{2}4x}\right)\)
30
Subtract the two binomials.
\(\left({10x^{2}6x}\right)\left({8x^{2}10x}\right)\)
31
Subtract the two binomials.
\(\left({6x^{5}+9x^{4}}\right)\left({2x4}\right)\)
32
Subtract the two binomials.
\(\left({6x^{6}x^{5}}\right)\left({4x^{3}+9}\right)\)
33
Subtract the two polynomials.
\(\left({5x^{3}3x^{2}+6x+9}\right)\left({8x^{2}10x+(1)}\right)\)
34
Subtract the two polynomials.
\(\left({6x^{3}8x^{2}+7x+5}\right)\left({3x^{2}+2x+(7)}\right)\)
35
Subtract the two trinomials.
\(\left({7x^{2}+10x+5}\right)\left({10x^{2}+3x9}\right)\)
36
Subtract the two trinomials.
\(\left({8x^{2}5x+5}\right)\left({5x^{2}+6x3}\right)\)
37
Subtract the two trinomials, making sure to simplify your answer as much as possible.
\(\left({8t^{6}+2t^{4}+9t^{2}}\right)\left({3t^{6}+2t^{4}+10t^{2}}\right)\)
38
Subtract the two trinomials, making sure to simplify your answer as much as possible.
\(\left({5t^{6}+9t^{4}2t^{2}}\right)\left({3t^{6}5t^{4}+2t^{2}}\right)\)
39
Simplify the following expression
\(\left[{x^{5}5x^{3}+6x^{2}}  \left({4x^{5}+10x^{3}8x^{2}}\right)\right]  \left({4x^{5}4x^{3}3x^{2}}\right)\)
40
Simplify the following expression
\(\left[{8x^{18}10x^{12}+3x^{6}}  \left({4x^{18}+5x^{12}7x^{6}}\right)\right]  \left({9x^{18}10x^{12}8x^{6}}\right)\)
41
Simplify the following expression
\(\left[{4y^{16}+10y^{15}}  \left({4y^{16}6y^{15}}\right)\right]  \left[{5y^{16}+8y^{15}}+\left({6y^{16}7y^{15}}\right)\right]\)
42
Simplify the following expression
\(\left[{y^{17}3y^{15}+8y^{6}}  \left({5y^{17}+5y^{15}5y^{6}}\right)\right]  \left[{2y^{17}5y^{15}+3y^{6}}+\left({9y^{17}5y^{15}8y^{6}}\right)\right]\)
43
Subtract \(7y^{15}2y^{13}4y^{6}\) from the sum of \(7y^{15}8y^{13}+5y^{6}\) and \(5y^{15}+10y^{13}5y^{6}\text{.}\)
44
Subtract \(3r^{8}8r^{7}9r^{3}\) from the sum of \(4r^{8}4r^{7}+2r^{3}\) and \(5r^{8}+5r^{7}4r^{3}\text{.}\)
45
Add or subtract the given polynomials as indicated
\(\displaystyle{ \left({8x^{6}y^{7}+3xy}\right)+\left({7x^{6}y^{7}9xy}\right) }\)
46
Add or subtract the given polynomials as indicated
\(\displaystyle{ \left({9x^{8}y^{3}+6xy}\right)+\left({7x^{8}y^{3}+10xy}\right) }\)
47
Add or subtract the given polynomials as indicated
\(\displaystyle{ \left({10x^{8}y^{3}10xy3}\right)+\left({7x^{8}y^{3}2xy+4}\right) }\)
48
Add or subtract the given polynomials as indicated
\(\displaystyle{ \left({2x^{7}y^{3}5xy+6}\right)+\left({7x^{7}y^{3}2xy+10}\right) }\)
49
Add or subtract the given polynomials as indicated
\(\left({3x^{8}y^{6}+9x^{5}y^{4}8xy}\right)+\left({7x^{8}y^{6}+3x^{5}y^{4}6xy}\right)\)
50
Add or subtract the given polynomials as indicated
\(\left({4x^{7}y^{6}3x^{3}y^{4}+2xy}\right)+\left({7x^{7}y^{6}4x^{3}y^{4}3xy}\right)\)
51
Add or subtract the given polynomials as indicated
\(\left({5x^{4}7xy5y^{2}}\right)\left({6x^{4}+5xy9y^{2}}\right)\)
52
Add or subtract the given polynomials as indicated
\(\left({6x^{3}+2xy3y^{4}}\right)\left({7x^{3}5xy+7y^{4}}\right)\)
53
Add or subtract the given polynomials as indicated
\(\left({7x^{9}y^{8}+6x^{2}y^{3}2xy}\right)\left({6x^{9}y^{8}10x^{2}y^{3}4xy}\right)\)
54
Add or subtract the given polynomials as indicated
\(\left({8x^{9}y^{7}9x^{5}y^{3}+4xy}\right)\left({6x^{9}y^{7}+7x^{5}y^{3}8xy}\right)\)
55
Add or subtract the given polynomials as indicated
\(\left({9x^{8}4y^{5}}\right)\left({6x^{8}8x^{5}y^{5}7x^{8}y^{5}4y^{5}}\right)\)
56
Add or subtract the given polynomials as indicated
\(\left({10x^{7}+8y^{5}}\right)\left({6x^{7}+9x^{2}y^{5}+9x^{7}y^{5}10y^{5}}\right)\)
57
Subtract \({2x^{4}y^{3}+3xy}\) from \({6x^{4}y^{3}10xy}\)
58
Subtract \({3x^{5}y^{4}6xy}\) from \({6x^{5}y^{4}10xy}\)
Applications of Simplifying Polynomials
59
An auto company’s sales volume can be modeled by \({4.3x^{2}+5.7x+2.8}\text{,}\) and its cost can be modeled by \({2.7x^{2}+4.7x+2.8}\text{,}\) where \(x\) represents the number of cars produced, and \(y\) stands for money in thousand dollars. We can calculate the company’s net profit by subtracting cost from sales. Find the polynomial which models the company’s sales in thousands of dollars.
The company’s profit can be modeled by dollars.
60
An auto company’s sales volume can be modeled by \({8x^{2}+6.5x+2.7}\text{,}\) and its cost can be modeled by \({3.1x^{2}1.2x+2.7}\text{,}\) where \(x\) represents the number of cars produced, and \(y\) stands for money in thousand dollars. We can calculate the company’s net profit by subtracting cost from sales. Find the polynomial which models the company’s sales in thousands of dollars.
The company’s profit can be modeled by dollars.
61
A handyman is building two pig pens sharing the same side. Assume the length of the shared side is \(x\) meters. The cost of building one pen would be \({33x^{2}6x8.5}\) dollars, and the cost of building the other pen would be \({34x^{2}+6x5}\) dollars. What’s the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
62
A handyman is building two pig pens sharing the same side. Assume the length of the shared side is \(x\) meters. The cost of building one pen would be \({22.5x^{2}+6x10}\) dollars, and the cost of building the other pen would be \({37.5x^{2}6x33}\) dollars. What’s the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
63
A farmer is building fence around a triangular area. The cost of building the shortest side is \({45x}\) dollars, where \(x\) stands for the length of the side in feet. The cost of building the other two sides can be modeled by \({7x^{2}+1.5x+50}\) dollars and \({3x^{3}+3.5x+30}\) dollars, respectively. What’s the total cost of building fence for all three sides?
The cost of building fence for all three sides would be dollars.
64
A farmer is building fence around a triangular area. The cost of building the shortest side is \({45x}\) dollars, where \(x\) stands for the length of the side in feet. The cost of building the other two sides can be modeled by \({5x^{2}4.5x+45}\) dollars and \({3x^{3}1.5x+30}\) dollars, respectively. What’s the total cost of building fence for all three sides?
The cost of building fence for all three sides would be dollars.
65
An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial \({6x^{4}+5x^{2}0.5x}\text{,}\) and the area of the house’s base can be modeled by \({3x^{3}0.5x15}\text{.}\) The rest of the plot is the yard. What’s the yard’s area?
The area of the yard can be modeled by the polynomial .
66
An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial \({2x^{4}+16x^{2}+8x}\text{,}\) and the area of the house’s base can be modeled by \({3x^{3}+8x15}\text{.}\) The rest of the plot is the yard. What’s the yard’s area?
The area of the yard can be modeled by the polynomial .