AP®︎ Precalculus: Topic 1.5 Practice Test

Essential Knowledge: 1.5.A.1

If $a$ is a complex number and $p(a)=0$, then $a$ is called a zero of the polynomial function $p$, or a root of $p(x)=0$. If $a$ is a real number, then $(x-a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$.

Essential Knowledge: 1.5.A.2

If a linear factor $(x-a)$ is repeated $n$ times, the corresponding zero of the polynomial function has a multiplicity $n$. A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities.

Essential Knowledge: 1.5.A.3

If $a$ is a real zero of a polynomial function $p$, then the graph of $y=p(x)$ has an $x$-intercept at the point $(a,0)$. Consequently, real zeros of a polynomial can be endpoints for intervals satisfying polynomial inequalities.

Essential Knowledge: 1.5.A.4

If $a+bi$ is a non-real zero of a polynomial function $p$, then its conjugate $a-bi$ is also a zero of $p$.

Essential Knowledge: 1.5.A.5

If the real zero, $a$, of a polynomial function has even multiplicity, then the signs of the output values are the same for input values near $x=a$. For these polynomial functions, the graph will be tangent to the $x$-axis at $x=a$.

Essential Knowledge: 1.5.A.6

The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value $n$ for which the successive $n$th differences are constant.

Essential Knowledge: 1.5.B.1

An even function is graphically symmetric over the line $x=0$ and analytically has the property $f(-x)=f(x)$. If $n$ is even, then a polynomial of the form $p(x)=a_nx^n$, where $n\geq 1$ and $a_n\ne 0$, is an even function.

Essential Knowledge: 1.5.B.2

An odd function is graphically symmetric about the point $(0,0)$ and analytically has the property $f(-x)=-f(x)$. If $n$ is odd, then a polynomial of the form $p(x)=a_nx^n$, where $n\geq 1$ and $a_n\ne 0$, is an odd function.