Polynomial functions flashcards: zeros to graphs

Question 1

What is a polynomial function $f(x)$?

Accepted Answer

A function of the form $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$$ where $n$ is a nonnegative integer, $a_n
eq 0$, and coefficients $a_i$ are real numbers.

Question 2

What is the multiplicity of a zero, and how does it affect the graph at that intercept?

Accepted Answer

If $(x-c)^m$ is a factor and $m\ge 1$ is largest, then $c$ has multiplicity $m$. Odd $m$: graph crosses the $x$-axis; even $m$: graph touches/bounces off the $x$-axis (higher $m$ tends to look flatter).

Question 3

State the Factor Theorem and explain how it helps with graphing polynomials.

Accepted Answer

Factor Theorem: $(x-c)$ is a factor of $f(x)$ iff $f(c)=0$. For graphing: use real zeros (and multiplicities) to place and describe $x$-intercepts, use $f(0)$ for the $y$-intercept, and use end behavior from the leading term to sketch the overall shape.

Polynomial functions flashcards: zeros to graphs

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