Graphs of Polynomial

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What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, f(x) = 5x^4 + 2x^3 + 2x - 7 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) = 5x^4 + 2x^3 + 2x - 7, the degree is 4.

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How many solutions can a polynomial have?

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A polynomial of degree n can have at most n solutions (roots). For example, a polynomial of degree 4 can have at most 4 solutions.

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What is end behavior of a polynomial function?

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

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What does it mean for a polynomial to be factored?

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Factoring a polynomial means expressing it as a product of its factors. For example, f(x) = (x-5)^2(x-2)(x+3) is a factored form of a polynomial.

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What is the significance of multiplicity in polynomial roots?

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Multiplicity refers to the number of times a particular root appears in a polynomial. A root with an even multiplicity touches the x-axis, while a root with an odd multiplicity crosses the x-axis.

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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 f(x) = 5x^4 + 2x^3 + 2x - 7, the leading coefficient is 5.

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How do you determine the x-intercepts of a polynomial?

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To find the x-intercepts of a polynomial, set the polynomial equal to zero and solve for x. The solutions are the x-intercepts.

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

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A polynomial of degree n can have at most n-1 turning points. Turning points are where the graph changes direction.

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What is a root of a polynomial?

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A root of a polynomial is a value of x that makes the polynomial equal to zero. For example, if f(x) = 0, then x is a root.