9.1: Addition and Subtraction of Polynomials
So far we have discussed linear functions and exponential functions. This lesson introduces polynomial functions.
Definition: A polynomial is an expression made with constants, variables, and positive integer exponents of the variables.
An example of a polynomial is: \begin{align*}4x^3 + 2x^2  3x + 1\end{align*}
Example 1: Identify the following expressions as polynomials or nonpolynomials.
(a) \begin{align*}5x^2  2x\end{align*}
(b) \begin{align*}3x^2  2x^{2}\end{align*}
(c) \begin{align*}x\sqrt{x}  1\end{align*}
(d) \begin{align*}\frac{5}{x^3 + 1}\end{align*}
(e) \begin{align*}4x^{\frac{1}{3}}\end{align*}
(f) \begin{align*}4xy^2  2x^2y  3 + y^3  3x^3\end{align*}
Solution:
(a) \begin{align*}5x^2  2x\end{align*}
(b) \begin{align*}3x^2  2x^{2}\end{align*}
(c) \begin{align*}x\sqrt{x}  1\end{align*}
(d) \begin{align*}\frac{5}{x^3 + 1}\end{align*}
(e) \begin{align*}4x^{\frac{1}{3}}\end{align*}
(f) \begin{align*}4xy^2  2x^y  3 + y^3  3x^3\end{align*}
Classifying Polynomials by Degree
The degree of a polynomial is the largest exponent of a single term.

\begin{align*}4x^3\end{align*}
4x3 has a degree of 3 and is called a cubic term or \begin{align*}3^{rd}\end{align*}3rd order term. 
\begin{align*}2x^2\end{align*}
2x2 has a degree of 2 and is called a quadratic term or \begin{align*}2^{nd}\end{align*}2nd order term. 
\begin{align*}3x\end{align*}
−3x has a degree of 1 and is called a linear term or \begin{align*}1^{st}\end{align*}1st order term.  1 has a degree of 0 because there is no variable.
Polynomials can have more than one variable. Here is another example of a polynomial: \begin{align*}t^46s^3t^212st+4s^45\end{align*}
\begin{align*}t^4\end{align*}
\begin{align*}6s^3t^2\end{align*}
\begin{align*}12^{st}\end{align*}
\begin{align*}4s^4\end{align*}
–5 is a constant, so its degree is 0.
Since the highest degree of a term in this polynomial is 5, this is a polynomial of degree 5 or a \begin{align*}5^{th}\end{align*}
Example 2: Identify the coefficient on each term, the degree of each term, and the degree of the polynomial.
\begin{align*}x^43x^3y^2+8x12\end{align*}
Solution: The coefficients of each term in order are 1, –3, 8 and the constant is –12.
The degrees of each term are 4, 5, 1, and 0. Therefore, the degree of the polynomial is 5.
A monomial is a onetermed polynomial. It can be a constant, a variable, or a combination of constants and variables. Examples of monomials are: \begin{align*}b^2; \ 6; \ 2ab^2; \ \frac{1}{4} x^2\end{align*}
Rewriting Polynomials in Standard Form
Often, we arrange the terms in a polynomial in standard from in which the term with the highest degree is first and is followed by the other terms in order of decreasing power. The first term of a polynomial in this form is called the leading term, and the coefficient in this term is called the leading coefficient.
Example 3: Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.
(a) \begin{align*}73x^3+4x\end{align*}
(b) \begin{align*}aba^3+2b\end{align*}
Solution:
(a) \begin{align*}73x^3+4x\end{align*} is rearranged as \begin{align*}3x^3+4x+7\end{align*}. The leading term is \begin{align*}3x^3\end{align*} and the leading coefficient is –3.
(b) \begin{align*}aba^3+2b\end{align*} is rearranged as \begin{align*}a^3+ab+2b\end{align*}. The leading term is \begin{align*}a^3\end{align*} and the leading coefficient is –1.
Simplifying Polynomials
A polynomial is simplified if it has no terms that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.
\begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms.
\begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.
If we have a polynomial that has like terms, we simplify by combining them.
\begin{align*}& x^2 + \underline{6xy}\underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}
This polynomial is simplified by combining the like terms \begin{align*}6xy4xy=2xy\end{align*}. We write the simplified polynomial as \begin{align*}x^2+2xy+y^2\end{align*}.
Example 4: Simplify by collecting and combining like terms.
\begin{align*}a^3b^3  5ab^4 + 2a^3b  a^3b^3 + 3ab^4  a^2b\end{align*}
Solution: Use the Commutative Property of Addition to reorganize like terms then simplify.
\begin{align*}& = (a^3b^3a^3b^3) + (5ab^4+3ab^4) + 2a^3ba^2b\\ & = 02ab^4+2a^3ba^2b\\ & = 2ab^4+2a^3 ba^2 b\end{align*}
Adding and Subtracting Polynomials
To add or subtract polynomials, you have to group the like terms together and combine them to simplify.
Example 5: Add and simplify \begin{align*}3x^24x+7\end{align*} and \begin{align*}2x^34x^26x+5\end{align*}.
Solution: Add \begin{align*}3x^24x+7\end{align*} and \begin{align*}2x^34x^26x+5\end{align*}.
\begin{align*}(3x^24x+7)+(2x^34x^26x+5)&=2x^3+(3x^24x^2 )+(4x6x)+(7+5)\\ &=2x^3x^210x+12\end{align*}
Multimedia Link: For more explanation of polynomials, visit http://www.purplemath.com/modules/polydefs.htm  Purplemath’s website.
Example 6: Subtract \begin{align*}5b^22a^2\end{align*} from \begin{align*}4a^28ab9b^2\end{align*}.
Solution:
\begin{align*}(4a^28ab9b^2)(5b^22a^2)&=[(4a^2 (2a^2)]+(9b^25b^2)8ab\\ & = 6a^214b^28ab\end{align*}
Solving RealWorld Problems Using Addition or Subtraction of Polynomials
Polynomials are useful for finding the areas of geometric objects. In the following examples, you will see this usefulness in action.
Example 7: Write a polynomial that represents the area of each figure shown.
(a)
(b)
Solution: The blue square has area: \begin{align*}y \cdot y=y^2\end{align*}.
The yellow square has area: \begin{align*}x \cdot x = x^2\end{align*}.
The pink rectangles each have area: \begin{align*}x \cdot y =xy\end{align*}.
\begin{align*}\text{Test area} & = y^2+x^2+xy+xy\\ & = y^2 + x^2 + 2xy\end{align*}
To find the area of the green region we find the area of the big square and subtract the area of the little square.
The big square has area \begin{align*}y \cdot y =y^2\end{align*}.
The little square has area \begin{align*}x \cdot x = x^2\end{align*}.
Area of the green region \begin{align*}= y^2x^2\end{align*}
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Addition and Subtraction of Polynomials (15:59)
Define the following key terms.
 Polynomial
 Monomial
 Degree
 Leading coefficient
For each of the following expressions, decide whether it is a polynomial. Explain your answer.
 \begin{align*}x^2+3x^{\frac{1}{2}}\end{align*}
 \begin{align*}\frac{1}{3}x^2y9y^2\end{align*}
 \begin{align*}3x^{3}\end{align*}
 \begin{align*}\frac{2}{3}t^2\frac{1}{t^2}\end{align*}
Express each polynomial in standard form. Give the degree of each polynomial.
 \begin{align*}32x\end{align*}
 \begin{align*}8x^4x+5x^2+11x^410\end{align*}
 \begin{align*}84x+3x^3\end{align*}
 \begin{align*}16+5f^87f^3\end{align*}
 \begin{align*}5+2x5x^2+8x^3\end{align*}
 \begin{align*}x^29x^4+12\end{align*}
Add and simplify.
 \begin{align*}(x+8)+(3x5)\end{align*}
 \begin{align*}(8r^46r^23r+9)+(3r^3+5r^2+12r9)\end{align*}
 \begin{align*}(2x^2+4x12) + (7x+x^2)\end{align*}
 \begin{align*}(2a^2b2a+9)+(5a^2b4b+5)\end{align*}
 \begin{align*}(6.9a^22.3b^2+2ab)+(3.1a2.5b^2+b)\end{align*}
Subtract and simplify.
 \begin{align*}(t+15t^2)(5t^2+2t9)\end{align*}
 \begin{align*}(y^2+4y5)(5y^2+2y+7)\end{align*}
 \begin{align*}(h^7+2h^5+13h^3+4h^2h1)(3h^5+20h^33h^2+8h4)\end{align*}
 \begin{align*}(5m^2m)(3m^2+4m5)\end{align*}
 \begin{align*}(2a^2b3ab^2+5a^2b^2)(2a^2b^2+4a^2b5b^2)\end{align*}
Find the area of the following figures.
Mixed Review
 Solve by graphing \begin{align*}\begin{cases} y=\frac{1}{3} x4\\ y=4x+10 \end{cases}\end{align*}.
 Solve for \begin{align*}u\end{align*}: \begin{align*}12= \frac{4}{u}\end{align*}.
 Graph \begin{align*}y=x4+3\end{align*} on a coordinate plane.
 State its domain and range.
 How has this graph been shifted from the parent function \begin{align*}f(x)=x\end{align*}?
 Two dice are rolled. The sum of the values are recorded.
 Define the sample space.
 What is the probability the sum of the dice is nine?
 Consider the equation \begin{align*}y=6500(0.8)^x\end{align*}.
 Sketch the graph of this function.
 Is this exponential growth or decay?
 What is the initial value?
 What is its domain and range?
 What is the value when \begin{align*}x=9.5\end{align*}?
 Write an equation for the line that is perpendicular to \begin{align*}y=5\end{align*} and contains the ordered pair (6, –5)