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 ≠ 0.

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What are the degrees of polynomial functions?

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The degree of a polynomial function is the highest power of the variable in the expression. For example, in f(x) = 2x^3 + 3x^2 - x + 5, the degree is 3.

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What is the end behavior of polynomial functions?

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The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term.

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What does it mean for a polynomial to be even?

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A polynomial is even if f(-x) = f(x) for all x in its domain. This means the graph is symmetric with respect to the y-axis.

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What does it mean for a polynomial to be odd?

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A polynomial is odd if f(-x) = -f(x) for all x in its domain. This means the graph is symmetric with respect to the origin.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the function equal to zero and solve for x. The solutions are the x-intercepts of the graph.

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

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

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient determines the direction of the graph's end behavior. If it is positive, the graph rises to the right; if negative, it falls to the right.

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How do you add polynomial functions?

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To add polynomial functions, combine like terms by adding their coefficients. For example, (3x^2 + 2x) + (4x^2 - x) = 7x^2 + x.

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What is the process for multiplying polynomial functions?

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To multiply polynomials, use the distributive property (FOIL for binomials) to multiply each term in the first polynomial by each term in the second.