Using the drop-down selectors, classify the polynomial and identify its leading coefficient and constant.
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In the tables below, hover over any example to see a tooltip with the leading coefficient and constant term. Try to guess first!
Classification by Degree: The degree of a polynomial is its largest exponent. The degree will determine many properties of a polynomial, including how many roots it has and the shape of its graph. Thus it is helpful to name the common degrees.
Name | Degree | Standard Form | Example | Notes |
---|---|---|---|---|
Constant | 0 | a | -6 | A constant expression is non-variable, so it has no coefficients. You could say the leading coefficient is 0. |
Linear | 1 | ax+b | 4x-6 | |
Quadratic | 2 | ax^{2}+bx+c | 3x^{2}+4x-6 | Why "quadratic" when quad means 4? Think of a square. It has 4 sides, but the expression of its area is represented by x^{2}, or one side of the square raised to the second power. |
Cubic | 3 | ax^{3}+bx^{2}+cx+d | 3x^{2}-x^{3}-6+4x | This example is not in standard form. It is still cubic, with a leading coefficient of -1 and a constant term of -6 |
Quartic | 4 | ax^{4}+bx^{3}+cx^{2}+dx+e | x^{4}-x^{3}+3x^{2}+4x-6 | In this example, the leading coefficient is the implied 1. |
Quintic | 5 | ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f | x^{5}-x^{3} | In this example, the missing constant term is considered 0. |
Classification by Terms: The number of terms a polynomial has may be helpful when determining appropriate methods of factoring and solving. Special techniques may be applied to 1-, 2-, or 3-term polynomials, so they are commonly referred by this classification.
Name | Number of Terms | Example | Notes |
---|---|---|---|
Monomial | 1 | -5x^{3} | Any degree polynomial may be a monomial, but a constant term MUST be a monomial. |
Binomial | 2 | x^{3}-27 | Linear expressions can only be binomial or monomial. |
Trinomial | 3 | 5x^{2}-3x-2 | Quadratic expressions can be trinomial or smaller. |
Polynomial | 4 or more | x^{4}-x^{3}+3x^{2}+4x-6 | Can you infer the maximum number of terms given a polynomial's degree? |