Polynomial Functions & their 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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n$$ is the leading coefficient and $$n$$ is a non-negative integer.

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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 the polynomial $$3x^4 + 2x^3 - x + 5$$, the degree is 4.

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How do you determine the number of possible zeros of a polynomial?

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

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What is a leading coefficient?

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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 $$4x^3 + 2x^2 - x + 7$$, the leading coefficient is 4.

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

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The maximum number of turns in the graph of a polynomial function is one less than its degree. For example, a polynomial of degree 3 can have at most 2 turns.

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What is end behavior in polynomial functions?

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

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Describe the end behavior of a polynomial with an even degree and a positive leading coefficient.

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The end behavior will be: Up, Up (as x approaches ±∞, f(x) approaches +∞).

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Describe the end behavior of a polynomial with an odd degree and a negative leading coefficient.

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The end behavior will be: Up, Down (as x approaches -∞, f(x) approaches +∞ and as x approaches +∞, f(x) approaches -∞).

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What is a zero of a polynomial function?

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A zero of a polynomial function is a value of x for which the polynomial evaluates to zero. It represents the x-intercepts of the graph.

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How can you find the zeros of a polynomial function?

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Zeros can be found by factoring the polynomial, using the Rational Root Theorem, or applying numerical methods such as synthetic division or the quadratic formula.