Polynomial Graphs

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients. Example: f(x) = 2x^3 - 4x^2 + 3x - 5.

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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. Example: In f(x) = 3x^4 + 2x^3 - x + 7, the degree is 4.

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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's graph.

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How do you find the y-intercept of a polynomial function?

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To find the y-intercept, evaluate the polynomial at x = 0. Example: For f(x) = x^2 + 3x + 2, the y-intercept is f(0) = 2.

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What are the zeros of a polynomial?

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The zeros of a polynomial are the values of x for which the polynomial equals zero. They are also known as roots.

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What is the multiplicity of a zero?

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The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. Example: If (x - 2)^3 is a factor, then 2 has a multiplicity of 3.

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What is the maximum number of turns a polynomial graph can have?

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A polynomial of degree n can have at most n - 1 turns.

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How does the sign of the leading coefficient affect the graph of a polynomial?

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If the leading coefficient is positive, the graph rises to the right; if negative, it falls to the right.

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What is the end behavior of a polynomial function?

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The end behavior describes how the graph behaves as x approaches positive or negative infinity, determined by the degree and leading coefficient.

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What is a factored form of a polynomial?

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Factored form expresses a polynomial as a product of its factors. Example: f(x) = (x - 1)(x + 2) is a factored form.