Intermediate Algebra

Module 8: Polynomials and Polynomial Functions

Read: Define and Evaluate Polynomials

Learning Objectives

  • Identify the degree and leading coefficient of a polynomial
  • Evaluate a polynomial for given values


In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a polynomial.

A polynomial function is a function consisting of sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variables with an non-negative integer exponents. Non negative integers are 
0,1,2,3,40, 1, 2, 3, 4
, ...

You may see a resemblance between expressions and polynomials, which we have been studying in this course.  Polynomials are a special sub-group of mathematical expressions and equations.

The following table is intended to help you tell the difference between what is a polynomial and what is not.

IS a Polynomial Is NOT a Polynomial Because
2x212x92x^2-\frac{1}{2}x -9
2x2+x\frac{2}{x^{2}}+x
Polynomials only have variables in the numerator
y4y3\frac{y}{4}-y^3
2y+4\frac{2}{y}+4
Polynomials only have variables in the numerator
12(a)+9\sqrt{12}\left(a\right)+9
 
a+7\sqrt{a}+7
Roots are equivalent to rational exponents, and polynomials only have integer exponents
The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.

The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent. A polynomial containing two terms, such as
2x92x - 9
, is called a binomial. A polynomial containing three terms, such as
3x2+8x7-3{x}^{2}+8x - 7
, is called a trinomial.

We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.

4x^3 - 9x^2 + 6x, with the text

Given a polynomial expression, identify the degree and leading coefficient.

  1. Find the highest power of x to determine the degree.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.


Example

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

  1. 3+2x24x33+2{x}^{2}-4{x}^{3}
  2. 5t52t3+7t5{t}^{5}-2{t}^{3}+7t
  3. 6pp326p-{p}^{3}-2






 

In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial.



The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.

Monomials Binomials Trinomials Other Polynomials
1515
3y+133y+13
x3x2+1x^{3}-x^{2}+1
5x4+3x36x2+2x5x^{4}+3x^{3}-6x^{2}+2x
12x \displaystyle \frac{1}{2}x
4p74p-7
3x2+2x93x^{2}+2x-9
13x52x4+29x3x2+4x56\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}
4y3-4y^{3}
3x2+58x3x^{2}+\frac{5}{8}x
3y3+y223y^{3}+y^{2}-2
3t33t23t33t^{3}-3t^{2}-3t-3
16n416n^{4}
14y3+3y14y^{3}+3y
a7+2a53a3a^{7}+2a^{5}-3a^{3}
q7+2q53q3+qq^{7}+2q^{5}-3q^{3}+q
When the coefficient of a polynomial term is 
00
, you usually do not write the term at all (because 
00
times anything is 
00
, and adding 
00
doesn’t change the value). The last binomial above could be written as a trinomial,
14y3+0y2+3y14y^{3}+0y^{2}+3y
.

A term without a variable is called a constant term, and the degree of that term is 
00
. For example 
1313
is the constant term in
3y+133y+13
. You would usually say that
14y3+3y14y^{3}+3y
has no constant term or that the constant term is 
00
.

Evaluate a polynomial

You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.

Example

Evaluate
3x22x+13x^{2}-2x+1
for
x=1x=-1
.





Example

Evaluate
23p4+23p \displaystyle -\frac{2}{3}p^{4}+2^{3}-p
for
p=3p = 3
.





 IN the following video we show more examples of evaluating polynomials for given values of the variable.



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