Graphing Polynomials

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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 - 3x^2 + 4x - 10 is a polynomial.

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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 f(x) = 2x^3 - 3x^2 + 4x - 10, the degree is 3.

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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 that make the polynomial equal to zero. They are also known as roots.

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. This may involve factoring, using the quadratic formula, or other methods.

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What is multiplicity in relation to polynomial zeros?

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Multiplicity refers to the number of times a particular zero is repeated in a polynomial. For example, if (x - 2) is squared in the polynomial, then 2 is a zero with multiplicity 2.

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

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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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What does the graph of a polynomial function look like?

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The graph of a polynomial function is a smooth, continuous curve that can have various shapes depending on the degree and leading coefficient.

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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 up to n real zeros, counting multiplicities.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by the leading term. If the leading coefficient is positive, the ends of the graph will rise; if negative, the ends will fall.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor, which is faster than long division.