Read: Define and Evaluate Polynomials
Learning Objectives
- Identify the degree and leading coefficient of a polynomial
- Evaluate a polynomial for given values
IS a Polynomial | Is NOT a Polynomial | Because |
[latex]2x^2-\frac{1}{2}x -9[/latex] | [latex]\frac{2}{x^{2}}+x[/latex] | Polynomials only have variables in the numerator |
[latex]\frac{y}{4}-y^3[/latex] | [latex]\frac{2}{y}+4[/latex] | Polynomials only have variables in the numerator |
[latex]\sqrt{12}\left(a\right)+9[/latex] | [latex]\sqrt{a}+7[/latex] | Roots are equivalent to rational exponents, and polynomials only have integer exponents |
Given a polynomial expression, identify the degree and leading coefficient.
- Find the highest power of x to determine the degree.
- Identify the term containing the highest power of x to find the leading term.
- Identify the coefficient of the leading term.
Example
For the following polynomials, identify the degree, the leading term, and the leading coefficient.- [latex]3+2{x}^{2}-4{x}^{3}[/latex]
- [latex]5{t}^{5}-2{t}^{3}+7t[/latex]
- [latex]6p-{p}^{3}-2[/latex]
Answer:
- The highest power of x is [latex]3[/latex], so the degree is [latex]3[/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[/latex]. The leading coefficient is the coefficient of that term, [latex]-4[/latex].
- The highest power of t is [latex]5[/latex], so the degree is [latex]5[/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[/latex]. The leading coefficient is the coefficient of that term, [latex]5[/latex].
- The highest power of p is [latex]3[/latex], so the degree is [latex]3[/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[/latex], The leading coefficient is the coefficient of that term, [latex]-1[/latex].
Monomials | Binomials | Trinomials | Other Polynomials |
[latex]15[/latex] | [latex]3y+13[/latex] | [latex]x^{3}-x^{2}+1[/latex] | [latex]5x^{4}+3x^{3}-6x^{2}+2x[/latex] |
[latex] \displaystyle \frac{1}{2}x[/latex] | [latex]4p-7[/latex] | [latex]3x^{2}+2x-9[/latex] | [latex]\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}[/latex] |
[latex]-4y^{3}[/latex] | [latex]3x^{2}+\frac{5}{8}x[/latex] | [latex]3y^{3}+y^{2}-2[/latex] | [latex]3t^{3}-3t^{2}-3t-3[/latex] |
[latex]16n^{4}[/latex] | [latex]14y^{3}+3y[/latex] | [latex]a^{7}+2a^{5}-3a^{3}[/latex] | [latex]q^{7}+2q^{5}-3q^{3}+q[/latex] |
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 [latex]3x^{2}-2x+1[/latex] for [latex]x=-1[/latex].Answer: Substitute [latex]-1[/latex] for each x in the polynomial.
[latex]3\left(-1\right)^{2}-2\left(-1\right)+1[/latex]
Following the order of operations, evaluate exponents first.[latex]3\left(1\right)-2\left(-1\right)+1[/latex]
Multiply [latex]3[/latex] times [latex]1[/latex], and then multiply [latex]-2[/latex] times [latex]-1[/latex].[latex]3+\left(-2\right)\left(-1\right)+1[/latex]
Change the subtraction to addition of the opposite.[latex]3+2+1[/latex]
Find the sum.Answer
[latex-display]3x^{2}-2x+1=6[/latex], for [latex]x=-1[/latex-display]Example
Evaluate [latex] \displaystyle -\frac{2}{3}p^{4}+2^{3}-p[/latex] for [latex]p = 3[/latex].Answer: Substitute [latex]3[/latex] for each p in the polynomial.
[latex] \displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-3[/latex]
Following the order of operations, evaluate exponents first and then multiply.[latex] \displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3[/latex]
Add and then subtract to get [latex]-3[/latex].[latex]-54 + 54 – 3[/latex]
Answer
[latex-display] \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3[/latex], for [latex]p = 3[/latex-display]IN the following video we show more examples of evaluating polynomials for given values of the variable.
https://youtu.be/2EeFrgQP1hMLicenses & Attributions
CC licensed content, Original
- Evaluate a Polynomial in One Variable. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.
- College Algebra. Authored by: Abramson, Jay, et al.. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: Public Domain: No Known Copyright. License terms: Download for free at :http://cnx.org/contents/[email protected]:1/Preface.
- Ex: Intro to Polynomials in One Variable. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.