Polynomial Graphs

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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 $$a_n$$ is not zero.

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What are the zeros of a polynomial function?

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

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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 $$x$$. This may involve factoring, using the quadratic formula, or other algebraic methods.

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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 in a polynomial. It determines the end behavior of the graph of the polynomial function.

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What does the degree of a polynomial indicate?

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The degree of a polynomial is the highest power of the variable in the polynomial. It indicates the number of roots (including multiplicities) and the general shape of the graph.

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What is the FOIL method?

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The FOIL method is a technique for multiplying two binomials. It stands for First, Outside, Inside, Last, referring to the order in which you multiply the terms.

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How do you create a polynomial from its zeros?

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To create a polynomial from its zeros, write the factors corresponding to each zero and multiply them together. For example, if the zeros are $$r_1, r_2, r_3$$, the polynomial is $$f(x) = k(x - r_1)(x - r_2)(x - r_3)$$ where $$k$$ is a constant.

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What is the relationship between the degree of a polynomial and its graph?

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The degree of a polynomial affects the number of turning points and the end behavior of its graph. An odd degree polynomial has opposite end behaviors, while an even degree polynomial has the same end behavior.

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What are multiplicities in polynomial functions?

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Multiplicity refers to the number of times a particular zero appears in a polynomial. A zero with an odd multiplicity will cross the x-axis, while a zero with an even multiplicity will touch the x-axis and turn around.

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What is the difference between even and odd degree polynomials?

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Even degree polynomials have a symmetric graph about the y-axis, while odd degree polynomials have a graph that is symmetric about the origin.