Suppose that two square plots of land both have a side length of \begin{align*}x\end{align*} feet. For one of the plots of land, the length is increased by 2 feet, and the width is decreased by 3 feet. For the other plot of land, the length is increased by 3 feet, and the width is decreased by 4 feet. What is the difference in the resulting areas of the plots? Could you rewrite the expression representing the difference in standard form? In this Concept, you'll learn how to classify, simplify, and rewrite polynomials in standard form so that you can produce expressions such as the one asked for here.
Guidance
So far we have discussed linear functions and exponential functions. This Concept 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*}. There are four terms: \begin{align*}4x^3, \ 2x^2, \ 3x,\end{align*} and 1. The numbers appearing in each term in front of the variable are called the coefficients. 4, 2, and 3 are coefficients because those numbers are in front of a variable. The number appearing all by itself without a variable is called a constant. 1 is the constant because it is by itself.
Example A
Identify the following expressions as polynomials or non-polynomials.
(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*} This is a polynomial.
(b) \begin{align*}3x^2 - 2x^{-2}\end{align*} This is not a polynomial because it has a negative exponent.
(c) \begin{align*}x\sqrt{x} - 1\end{align*} This is not a polynomial because is has a square root.
(d) \begin{align*}\frac{5}{x^3 + 1}\end{align*} This is not a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator.
(e) \begin{align*}4x^{\frac{1}{3}}\end{align*} This is not a polynomial because it has a fractional exponent.
(f) \begin{align*}4xy^2 - 2x^y - 3 + y^3 - 3x^3\end{align*} This is a polynomial.
Classifying Polynomials by Degree
The degree of a polynomial is the largest exponent of a single term.
- \begin{align*}4x^3\end{align*} has a degree of 3 and is called a cubic term or \begin{align*}3^{rd}\end{align*} order term.
- \begin{align*}2x^2\end{align*} has a degree of 2 and is called a quadratic term or \begin{align*}2^{nd}\end{align*} order term.
- \begin{align*}-3x\end{align*} has a degree of 1 and is called a linear term or \begin{align*}1^{st}\end{align*} 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^4-6s^3t^2-12st+4s^4-5\end{align*}. This is a polynomial because all exponents on the variables are positive integers. This polynomial has five terms. Note: The degree of a term is the sum of the powers on each variable in the term.
\begin{align*}t^4\end{align*} has a degree of 4, so it’s a \begin{align*}4^{th}\end{align*} order term.
\begin{align*}-6s^3t^2\end{align*} has a degree of 5, so it’s a \begin{align*}5^{th}\end{align*} order term.
\begin{align*}-12^{st}\end{align*} has a degree of 2, so it’s a \begin{align*}2^{nd}\end{align*} order term.
\begin{align*}4s^4\end{align*} has a degree of 4, so it’s a \begin{align*}4^{th}\end{align*} order term.
–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*} order polynomial.
Example B
Identify the coefficient on each term, the degree of each term, and the degree of the polynomial.
\begin{align*}x^4-3x^3y^2+8x-12\end{align*}
Solution: The coefficients of each term in order are 1, –3, and 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 one-termed polynomial. It can be a constant, a variable, or a variable with a coefficient. Examples of monomials are the following: \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 form in which the term with the highest degree is first and is followed by the other terms in order of decreasing powers. 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 C
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*}7-3x^3+4x\end{align*}
(b) \begin{align*}ab-a^3+2b\end{align*}
Solution:
(a) \begin{align*}7-3x^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*}ab-a^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*}6xy-4xy=2xy\end{align*}. We write the simplified polynomial as \begin{align*}x^2+2xy+y^2\end{align*}.
Example D
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 and then simplify.
\begin{align*}& = (a^3b^3-a^3b^3) + (-5ab^4+3ab^4) + 2a^3b-a^2b\\ & = 0-2ab^4+2a^3b-a^2b\\ & = -2ab^4+2a^3 b-a^2 b\end{align*}
Video Review
-->
Guided Practice
Simplify and rewrite the following polynomial in standard form. State the degree of the polynomial.
\begin{align*}16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}
Solution:
Start by simplifying by combining like terms:
\begin{align*}16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2=(16x^2y^3-7x^2y^3)-3xy^5+(-2x^3y^2+2x^3y^2)+2xy=9x^2y^3-3xy^5+2xy\end{align*}
In order to rewrite in standard form, we need to determine the degree of each term. The first term has a degree of \begin{align*}2+3=5\end{align*}, the second term has a degree of \begin{align*}1+5=6\end{align*}, and the last term has a degree of \begin{align*}1+1=2\end{align*}. We will rewrite the terms in order from largest degree to smallest degree:
\begin{align*}-3xy^5+9x^2y^3+2xy\end{align*}
The degree of a polynomial is the largest degree of all the terms. In this case that is 6.
Explore More
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. CK-12 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^2y-9y^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*}3-2x\end{align*}
- \begin{align*}8x^4-x+5x^2+11x^4-10\end{align*}
- \begin{align*}8-4x+3x^3\end{align*}
- \begin{align*}-16+5f^8-7f^3\end{align*}
- \begin{align*}-5+2x-5x^2+8x^3\end{align*}
- \begin{align*}x^2-9x^4+12\end{align*}
Mixed Review
- Solve by graphing \begin{align*}\begin{cases} y=\frac{1}{3} x-4\\ 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=|x-4|+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).