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The Language of Algebra
To identify the parts of a polynomial, including terms, coefficients, and degree.
To classify polynomials by the number of terms.
\( \underbrace{\textcolor{blue}{5}x^2}_{\text{Term}} + \underbrace{\textcolor{green}{3}x}_{\text{Term}} - \underbrace{\textcolor{red}{7}}_{\text{Term}} \)
Term: Each part of a polynomial separated by + or - signs. This example has 3 terms.
\( \textcolor{blue}{5}x^2 + \textcolor{green}{3}x - \textcolor{red}{7} \)
Coefficient: The number in front of a variable.
The coefficients in this example are 5 and 3.
Constant: A term without a variable (just a number).
The constant in this example is -7.
Monomial: A polynomial with one term. (e.g., \(7\) or \(6x^2\))
Binomial: A polynomial with two terms. (e.g., \(3t-1\))
Trinomial: A polynomial with three terms. (e.g., \(2x^2+3x-7\))
Polynomial: Any expression with 4 or more terms.
Degree of a Term: The exponent on the variable in that term.
The term \(8x^\textcolor{red}{3}\) has a degree of 3.
The term \(5x\) (which is \(5x^\textcolor{red}{1}\)) has a degree of 1.
Degree of a Polynomial: The largest degree of any of its terms.
The polynomial \(2x^\textcolor{red}{5} + 9x^2 - 1\) has a degree of 5.
The polynomial \(3x - 4x^\textcolor{red}{2}\) has a degree of 2.
Standard Form: Writing the terms in order from highest to lowest degree (exponent).
Not Standard: \(-4 + 3x + 9x^2\)
Standard Form: \(9x^2 + 3x - 4\)
Leading Coefficient: Once in standard form, it's the coefficient of the first term.
For \(\textcolor{red}{9}x^2 + 3x - 4\), the leading coefficient is 9.
Let's identify all the parts of this polynomial: \(4x - 2x^3 + 8\)
Standard Form: \(-2x^3 + 4x + 8\)
Degree: 3 (from the \(x^3\) term)
Leading Coefficient: -2
Number of Terms: 3
Name by Terms: Trinomial
Coefficients: -2, 4
Constant: 8
Identify all the parts of this polynomial:
\(7 + 5x^2 - 9x\)
Standard Form: \(5x^2 - 9x + 7\)
Degree: 2
Leading Coefficient: 5
Name by Terms: Trinomial
Try these problems on your handout.
Green Level
For \(2x - 5\), name the coefficients and the constant.
Yellow Level
For \(4x^2 - x + 9\), find the degree, leading coefficient, and name by terms.
Red Level
Write \(8 - x^3 + 2x\) in standard form and find the leading coefficient.
On your handout or a piece of paper, identify the following for the polynomial below:
a) Standard Form
b) Degree
c) Leading Coefficient
d) Name by Terms