Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Graphs

• The graph of a constant function

:f(x) = a_0

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is a horizontal line with y-intercept a_0.

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• The graph of a degree 1 polynomial function (or linear function)

:f(x) = a_0 + a_1 x, where a_1 eq 0

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is an oblique line with y-intercept a_0 and slope a_1.

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• The graph of a degree 2 or higher polynomial function

:f(x) = a_0 + a_1 x + cdots + a_{n - 1} x^{n - 1} + a_n x^n, where a_n eq 0 and n geq 2

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is a continuous non-linear curve.

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The best way to analyze the graph of a degree 2 or higher polynomial function is by its end behavior, the number of x-intercepts and the number of turning points.

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End behavior

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There are four end behaviors which are direct results of whether a_n, the leading coefficient, is positive or negative and whether n, the degree of the polynomial, is even or odd.

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• If $a\_n$ is positive and $n$ is even, the right end of the polynomial is in quadrant I while the left end is in quadrant II.
• If $a\_n$ is negative and $n$ is even, the right end is in quadrant IV while the left end is in quadrant III.
• If $a\_n$ is positive and $n$ is odd, the right end is in quadrant I while the left end is in quadrant III.
• If $a\_n$ is negative and $n$ is odd, the right end is in quadrant IV while the left end is in quadrant II.

Number of x-intercepts

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From the Fundamental theorem of algebra, a polynomial of degree n has exactly n complex roots, which may or may not be real. Therefore, the number of x-intercepts can't exceed n. It also follows from the Fundamental Theorem of Algebra that the complex roots of a polynomial must exist in conjugate pairs. This implies that an even-degree polynomial can have no x-intercepts (because all its roots may be complex); an odd-degree polynomial, on the other hand, must have at least one x-intercept ,since any pairing of roots into conjugate pairs will necesarily leave at least one unpaired for odd n. These "unpaired" roots must therefore be real. For example, a degree 4 polynomial function can have 0, 1, 2, 3 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 2, 3, 4 or 5 x-intercepts.

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Number of turning points

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The number of turning points of an even-degree polynomial is any odd number less than the degree, while the number of turning points of an odd-degree polynomial is any even number less than the degree. For example, a degree 4 polynomial function can have 1 or 3 turning points whereas a degree 5 polynomial function can have 0, 2, or 4 turning points.

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The following are some examples of polynomials of low degree.

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