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 $$a_n \neq 0$$.

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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 $$f(x) = 4x^3 + 2x^2 - x + 7$$, the degree is 3.

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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 4 can have up to 4 possible zeros.

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

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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 (both ends of the graph rise as 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: Down, Up (the left end of the graph falls and the right end rises as x approaches ±∞).

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What is a real zero of a polynomial?

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

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How can you find the real zeros of a polynomial function?

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Real zeros can be found by factoring the polynomial, using the Rational Root Theorem, or applying numerical methods such as the Newton-Raphson method.