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 ≠ 0.

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What does the degree of a polynomial indicate?

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The degree of a polynomial indicates the highest power of the variable in the polynomial. It determines the polynomial's end behavior and the maximum number of x-intercepts.

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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. It influences the direction of the graph as x approaches positive or negative infinity.

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How do you find the x-intercepts of a polynomial function?

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To find the x-intercepts, set the polynomial equal to zero and solve for x. The solutions are the x-intercepts, where the graph crosses the x-axis.

Tags

CCSS.HSF-IF.C.7C

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What is the significance of 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 represent the x-intercepts of the graph and are crucial for graphing the polynomial.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, f(x) = (x - r1)(x - r2)...(x - rn) where r1, r2, ..., rn are the roots.

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What is the end behavior of a polynomial function?

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The end behavior describes how the values of the polynomial function behave as x approaches positive or negative infinity. It is determined by the degree and leading coefficient.

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How do you determine the multiplicity of a root?

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The multiplicity of a root refers to the number of times a particular root appears in the factored form of the polynomial. It affects the shape of the graph at the x-intercept.

Tags

CCSS.HSF-IF.C.7C

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What is the relationship between the degree of a polynomial and the number of x-intercepts?

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The degree of a polynomial indicates the maximum number of x-intercepts it can have. A polynomial of degree n can have up to n real x-intercepts.

Tags

CCSS.HSF-IF.C.7C

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What is the difference between even and odd degree polynomials?

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Even degree polynomials have symmetric end behavior (both ends go up or down), while odd degree polynomials have opposite end behavior (one end goes up and the other goes down).