Polynomial Functions

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$ where $$a_n$$ is not zero.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in $$f(x) = 2x^3 + 3x^2 + 1$$, the degree is 3.

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What does the leading coefficient of a polynomial indicate?

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The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial function.

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How does the degree of a polynomial affect its end behavior?

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If the degree is even, the ends of the graph will either both go up or both go down. If the degree is odd, one end will go up and the other will go down.

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What is the end behavior of a polynomial function with an even degree and a positive leading coefficient?

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As $$x \to -\infty$$, $$f(x) \to +\infty$$ and as $$x \to +\infty$$, $$f(x) \to +\infty$$.

Front

What is the end behavior of a polynomial function with an odd degree and a negative leading coefficient?

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As $$x \to -\infty$$, $$f(x) \to +\infty$$ and as $$x \to +\infty$$, $$f(x) \to -\infty$$.

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What is the significance of the roots of a polynomial function?

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The roots (or zeros) of a polynomial function are the values of $$x$$ for which $$f(x) = 0$$. They indicate where the graph intersects the x-axis.

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CCSS.HSF-IF.C.7C

Front

What is the difference between a polynomial function and a rational function?

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A polynomial function consists of terms with non-negative integer exponents, while a rational function is a ratio of two polynomials.

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What is a turning point in the context of polynomial functions?

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A turning point is a point on the graph where the function changes direction from increasing to decreasing or vice versa.

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How can you determine the maximum number of turning points in a polynomial function?

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The maximum number of turning points of a polynomial function is one less than its degree. For example, a cubic polynomial can have at most 2 turning points.