2.3 Polynomials, End Behavior, Zeros, Multiplicities

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What is a polynomial function?

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A polynomial function is a mathematical expression that involves 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.

Front

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 the polynomial 2x^4 + 3x^2 + 1, 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 the polynomial 5x^3 + 2x^2 + 3, the leading coefficient is 5.

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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 the roots of the polynomial.

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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 appears in a polynomial. For example, if (x - 2) is a factor of a polynomial twice, then 2 is a zero with multiplicity 2.

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What is end behavior of a polynomial function?

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End behavior describes the behavior of the graph of a polynomial function as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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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 significance of the leading term in a polynomial?

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The leading term determines the end behavior of the polynomial function and influences the shape of the graph.

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What is a polynomial graph?

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A polynomial graph is the graphical representation of a polynomial function, showing its zeros, multiplicities, and end behavior.

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How can you identify a polynomial function from a graph?

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A polynomial function graph is continuous and smooth, with no breaks, sharp corners, or vertical asymptotes.