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 $$2x^3 + 3x^2 + 1$$, the degree is 3.

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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 the polynomial $$4x^5 + 2x^3 + 7$$, the leading coefficient is 4.

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What does it mean for a polynomial to be even?

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A polynomial is even if it is symmetric about the y-axis, meaning that $$f(-x) = f(x)$$ for all x. An example is $$f(x) = x^2$$.

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What does it mean for a polynomial to be odd?

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A polynomial is odd if it is symmetric about the origin, meaning that $$f(-x) = -f(x)$$ for all x. An example is $$f(x) = x^3$$.

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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. It is determined by the degree and leading coefficient.

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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 real zeros.

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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 4 can have up to 3 turns.

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What is standard form of a polynomial?

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The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$3x^4 - 2x^3 + x - 5$$ is in standard form.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or synthetic division.