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 does it mean for a polynomial to be in standard form?

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A polynomial is in standard form when its terms are arranged in descending order of their degree. For example, $$8x^3 - 4x^2 + 6x - 3$$ is in standard form.

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How can you determine if a polynomial function is even?

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A polynomial function is even if $f(-x) = f(x)$ for all $x$. This means that the graph is symmetric with respect to the y-axis.

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How can you determine if a polynomial function is odd?

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A polynomial function is odd if $f(-x) = -f(x)$ for all $x$. This means that the graph is symmetric with respect to the origin.

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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 $$8x^3 - 4x^2 + 6x - 3$$, 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 $$8x^3 - 4x^2 + 6x - 3$$, the leading coefficient is 8.

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What is the maximum number of real zeros a polynomial can have?

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The maximum number of real zeros a polynomial can have is equal to its degree. For example, a polynomial of degree 4 can have up to 4 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 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 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, i.e., $f(x) = 0$.

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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 numerical methods.