Sketching Polynomials Exit

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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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n$$ is not zero.

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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 expression. For example, in $$f(x) = 2x^3 + 3x^2 + 1$$, the degree is 3.

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How do you determine the number of roots of a polynomial?

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The number of roots of a polynomial is equal to its degree. For example, a polynomial of degree 3 can have up to 3 roots.

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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 graph.

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What does it mean for a polynomial to have a double root?

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A double root occurs when a root is repeated. For example, in $$f(x) = (x-2)^2$$, the root x=2 is a double root.

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What is the end behavior of a polynomial function?

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The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity, determined by the degree and leading coefficient.

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What is a factored form of a polynomial?

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A polynomial is in factored form when it is expressed as a product of its linear factors. For example, $$f(x) = (x-1)(x+2)$$ is a factored form.

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How can you identify the x-intercepts of a polynomial graph?

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The x-intercepts of a polynomial graph are the values of x for which the polynomial equals zero. They can be found by solving the equation $$f(x) = 0$$.

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What is the relationship between the degree of a polynomial and its number of turning points?

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A polynomial of degree n can have at most n-1 turning points.

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What is the vertex of a polynomial function?

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The vertex is the highest or lowest point on the graph of a polynomial function, depending on whether it opens upwards or downwards.