Polynomials & Graphing Review

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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. For example, f(x) = 2x^3 - 4x^2 + 3x - 5 is a polynomial.

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What does it mean for a zero to have multiplicity?

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Multiplicity refers to the number of times a particular root (or zero) appears in a polynomial. For example, in the polynomial (x - 1)^2, the zero x = 1 has a multiplicity of 2.

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What is the significance of the x-axis in graphing polynomials?

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The x-axis represents the values of x for which the polynomial equals zero. Points where the graph intersects or touches the x-axis are called zeros or roots of the polynomial.

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What does it mean for a zero to 'bounce' off the x-axis?

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A zero 'bounces' off the x-axis when it has an even multiplicity. This means the graph touches the x-axis at that point but does not cross it.

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What does it mean for a zero to 'cross' the x-axis?

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A zero 'crosses' the x-axis when it has an odd multiplicity. This means the graph intersects the x-axis at that point.

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How do you find the zeros of a polynomial in factored form?

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To find the zeros of a polynomial in factored form, set each factor equal to zero and solve for x. For example, for f(x) = (x - 2)(x + 3), the zeros are x = 2 and x = -3.

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What is the general form of a polynomial?

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The general form of a polynomial is given by P(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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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. For example, in the polynomial 3x^4 - 2x^2 + 5, the degree is 4.

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What is the leading coefficient of a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in 4x^3 + 2x^2 - 5, the leading coefficient is 4.

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What is the relationship between the degree of a polynomial and the number of zeros?

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A polynomial of degree n can have at most n real zeros, counting multiplicities.