08/14 8. Higher-Degree Polynomial Functions Free Online FlashCards

What defines a polynomial equation and its degree?

A polynomial equation has the form $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is the degree and $a_n\ne0$.

What does the Factor Theorem establish?

The Factor Theorem states that $f(k)=0$ if and only if $(x-k)$ is a factor of $f(x)$.

What does the Fundamental Theorem of Algebra guarantee?

Every nonconstant polynomial of degree $n$ has exactly $n$ complex zeros, counting multiplicity.

What is the Complex Conjugate Theorem?

If a polynomial has real coefficients and zero $a+bi$, it also has zero $a-bi$.

What candidates does the Rational Root Theorem identify?

For integer-coefficient $f(x)$, any rational zero $\frac{p}{q}$ in lowest terms has $p$ dividing the constant term and $q$ dividing the leading coefficient.

What continuity property does every polynomial graph have?

A polynomial graph is continuous: it has no breaks, holes, or vertical asymptotes.

How does multiplicity affect an x-intercept?

A zero with odd multiplicity usually makes the graph cross the x-axis; a zero with even multiplicity makes it touch the axis and turn around.

What is the end behavior of an odd-degree polynomial with positive lead?

An odd-degree polynomial with a positive leading coefficient falls on the left and rises on the right: $f(x)\to-\infty$ as $x\to-\infty$, and $f(x)\to\infty$ as $x\to\infty$.

What are the maximum turning points of a degree-$n$ polynomial?

A degree-$n$ polynomial has at most $n-1$ turning points and at most $n$ real zeros.

When should inequality endpoints be included?

For $f(x)\ge0$ or $f(x)\le0$, include zeros in the solution; for strict inequalities, exclude the zeros.

How does multiplicity affect sign changes?

Crossing a zero of odd multiplicity changes the polynomial's sign; crossing a zero of even multiplicity does not.

What is $(3+2i)(3-2i)$?

Using $i^2=-1$, $(3+2i)(3-2i)=13$. The middle terms cancel, and $-4i^2=4$.