What if you were given an algebraic expression like \begin{align*}3x - 2x^2 + 5 - x + 6x^2\end{align*}? How could you simplify it and find its degree? After completing this Concept, you'll be able to combine like terms to simplify polynomial expressions like this one and classify them by degree.
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CK-12 Foundation: 0901S Lesson Polynomial Expressions
Guidance
So far we’ve seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we’ll introduce polynomial functions. A polynomial is made up of different terms that contain positive integer powers of the variables. Here is an example of a polynomial:
@$$\begin{align*}4x^3+2x^2-3x+1\end{align*}@$$
Each part of the polynomial that is added or subtracted is called a term of the polynomial. The example above is a polynomial with four terms.
The numbers appearing in each term in front of the variable are called the coefficients. The number appearing all by itself without a variable is called a constant.
In this case the coefficient of @$\begin{align*}x^3\end{align*}@$ is 4, the coefficient of @$\begin{align*}x^2\end{align*}@$ is 2, the coefficient of @$\begin{align*}x\end{align*}@$ is -3 and the constant is 1.
Degrees of Polynomials and Standard Form
Each term in the polynomial has a different degree. The degree of the term is the power of the variable in that term.
@$$\begin{align*}& 4x^3 && \text{has degree} \ 3 \ \text{and is called a cubic term or} \ 3^{rd} \ \text{order term}.\\ & 2x^2 && \text{has degree} \ 2 \ \text{and is called a quadratic term or} \ 2^{nd} \ \text{order term}.\\ & -3x && \text{has degree} \ 1 \ \text{and is called a linear term or} \ 1^{st} \ \text{order term}.\\ & 1 && \text{has degree} \ 0 \ \text{and is called the constant}.\end{align*}@$$
By definition, the degree of the polynomial is the same as the degree of the term with the highest degree. This example is a polynomial of degree 3, which is also called a “cubic” polynomial. (Why do you think it is called a cubic?).
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 the exponents on the variables are positive integers. This polynomial has five terms. Let’s look at each term more closely.
Note: The degree of a term is the sum of the powers on each variable in the term. In other words, the degree of each term is the number of variables that are multiplied together in that term, whether those variables are the same or different.
@$$\begin{align*}& t^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}\\ & -6s^3t^2 && \text{has a degree of} \ 5, \ \text{so it's a} \ 5^{th} \ \text{order term}.\\ & -12st && \text{has a degree of} \ 2, \ \text{so it's a} \ 2^{nd} \ \text{order term}.\\ & 4s^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}.\\ & -5 && \text{is a constant, so its degree is} \ 0.\end{align*}@$$
Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree @$\begin{align*}5^{th}\end{align*}@$ or a @$\begin{align*}5^{th}\end{align*}@$ order polynomial.
A polynomial that has only one term has a special name. It is called a monomial (mono means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial, so a polynomial is just the sum of several monomials. Here are some examples of monomials:
@$$\begin{align*}b^2 \qquad -2ab^2 \qquad 8 \qquad \frac{1}{4}x^4 \qquad -29xy\end{align*}@$$
Example A
For the following polynomials, identify the coefficient of each term, the constant, the degree of each term and the degree of the polynomial.
a) @$\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}@$
b) @$\begin{align*}x^4-3x^3y^2+8x-12\end{align*}@$
Solution
a) @$\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}@$
The coefficients of each term in order are 1, -3, 4, and -5 and the constant is 7.
The degrees of each term are 5, 3, 2, 1, and 0. Therefore the degree of the polynomial is 5.
b) @$\begin{align*}x^4-3x^3y^2+8x-12\end{align*}@$
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.
Example B
Identify the following expressions as polynomials or non-polynomials.
a) @$\begin{align*}5x^5-2x\end{align*}@$
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) This is a polynomial.
b) This is not a polynomial because it has a negative exponent.
c) This is not a polynomial because it has a radical.
d) This is not a polynomial because the power of @$\begin{align*}x\end{align*}@$ appears in the denominator of a fraction (and there is no way to rewrite it so that it does not).
e) This is not a polynomial because it has a fractional exponent.
f) This is a polynomial.
Often, we arrange the terms in a polynomial in order of decreasing power. This is called standard form.
The following polynomials are in standard form:
@$$\begin{align*}4x^4-3x^3+2x^2-x+1\end{align*}@$$
@$$\begin{align*}a^4b^3-2a^3b^3+3a^4b-5ab^2+2\end{align*}@$$
The first term of a polynomial in standard form is called the leading term, and the coefficient of the leading term is called the leading coefficient.
The first polynomial above has the leading term @$\begin{align*}4x^4,\end{align*}@$ and the leading coefficient is 4.
The second polynomial above has the leading term @$\begin{align*}a^4b^3,\end{align*}@$ and the leading coefficient is 1.
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) @$\begin{align*}ab-a^3+2b\end{align*}@$
c) @$\begin{align*}-4b+4+b^2\end{align*}@$
Solution
a) @$\begin{align*}7-3x^3+4x\end{align*}@$ becomes @$\begin{align*}-3x^3+4x+7\end{align*}@$. Leading term is @$\begin{align*}-3x^3\end{align*}@$; leading coefficient is -3.
b) @$\begin{align*}ab-a^3+2b\end{align*}@$ becomes @$\begin{align*}-a^3+ab+2b\end{align*}@$. Leading term is @$\begin{align*}-a^3\end{align*}@$; leading coefficient is -1.
c) @$\begin{align*}-4b+4+b^2\end{align*}@$ becomes @$\begin{align*}b^2-4b+4\end{align*}@$. Leading term is @$\begin{align*}b^2\end{align*}@$; 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, whether they have the same or different coefficients.
For example, @$\begin{align*}2x^2y\end{align*}@$ and @$\begin{align*}5x^2y\end{align*}@$ are like terms, but @$\begin{align*}6x^2y\end{align*}@$ and @$\begin{align*}6xy^2\end{align*}@$ are not like terms.
When a polynomial has like terms, we can simplify it by combining those terms.
@$$\begin{align*}& x^2+\underline{6xy} - \underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}@$$
We can simplify this polynomial by combining the like terms @$\begin{align*}6xy\end{align*}@$ and @$\begin{align*}-4xy\end{align*}@$ into @$\begin{align*}(6-4)xy\end{align*}@$, or @$\begin{align*}2xy\end{align*}@$. The new polynomial is @$\begin{align*}x^2+2xy+y^2\end{align*}@$.
Example D
Simplify the following polynomials by collecting like terms and combining them.
a) @$\begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}@$
b) @$\begin{align*}a^3b^3-5ab^4+2a^3b-a^3b^3+3ab^4-a^2b\end{align*}@$
Solution
a) Rearrange the terms so that like terms are grouped together: @$\begin{align*}(-4x^2+x^2)+(2x+4x)+(6-4)\end{align*}@$
Combine each set of like terms: @$\begin{align*}-3x^2+6x+2\end{align*}@$
b) Rearrange the terms so that like terms are grouped together: @$\begin{align*}(a^3b^3-a^3b^3)+(-5ab^4+3ab^4)+2a^3b-a^2b\end{align*}@$
Combine each set of like terms: @$\begin{align*}0-2ab^4+2a^3b-a^2b=-2ab^4+2a^3b-a^2b\end{align*}@$
Watch this video for help with the Examples above.
CK-12 Foundation: Polynomial Expressions
Vocabulary
- A polynomial is an expression made with constants, variables, and positive integer exponents of the variables.
- In a polynomial, the number appearing in each term in front of the variables is called the coefficient.
- In a polynomial, the number appearing all by itself without a variable is called the constant.
- A monomial is a one-termed polynomial. It can be a constant, a variable, or a variable with a coefficient.
- The degree of a polynomial is the largest degree of the terms. The degree of a term is the power of the variable, or if the term has more than one variable, it is the sum of the powers on each variable.
- 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.
- Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.
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:
@$\begin{align*} 16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}@$
is equal to
@$\begin{align*}(16x^2y^3-7x^2y^3)-3xy^5+(-2x^3y^2+2x^3y^2)+2xy\end{align*}@$
which simplifies to
@$\begin{align*}9x^2y^3-3xy^5+2xy.\end{align*}@$
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
Indicate whether each expression is a polynomial.
- @$\begin{align*}x^2+3x^{\frac{1}{2}}\end{align*}@$
- @$\begin{align*}\frac{1}{3}x^2y-9y^2\end{align*}@$
- @$\begin{align*}3x^{-3}\end{align*}@$
- @$\begin{align*}\frac{2}{3}t^2-\frac{1}{t^2}\end{align*}@$
- @$\begin{align*}\sqrt{x}-2x\end{align*}@$
- @$\begin{align*}\left ( x^\frac{3}{2} \right )^2\end{align*}@$
Express each polynomial in standard form. Give the degree of each polynomial.
- @$\begin{align*}3-2x\end{align*}@$
- @$\begin{align*}8-4x+3x^3\end{align*}@$
- @$\begin{align*}-5+2x-5x^2+8x^3\end{align*}@$
- @$\begin{align*}x^2-9x^4+12\end{align*}@$
- @$\begin{align*}5x+2x^2-3x\end{align*}@$