Decide how zeros and end behavior of polynomial functions and their graphs are related to the degree and the factors of the polynomial.

A polynomial of degree n will have identically n complex zeros according to the Fundamental Theorem of Algebra. Any non-real complex zeros of the polynomial must appear in conjugate pairs if the polynomial's coefficients are all real (which is the case for the polynomials under discussion here).
A polynomial of degree 7, for instance, will either contain 0, 2, 4, or 6 non-real zeros, meaning it will either have 7, 5, 3, or 1 real zeros.
Take the polynomial as an illustration.
${\displaystyle f\left(x\right)=x^5-6x^4-3x^3+66x^2-58x-120}$
Graph has three real roots: ${\displaystyle x=-3,x=-1,x=4}$. The two remaining roots are complex. This means that we can write it as a product of factors where, for each real root ${\displaystyle a,x-a}$ will be a factor. Factor f as
${\displaystyle f\left(x\right)=\underset{realzeros}{\underbrace{(x-4)(x+1)(x+3)}}\underset{co\mp \le xzeros}{\underbrace{(x^2-6x+10)}}}$ and indeed, if solve ${\displaystyle x^2-6x+10=0}$, you will find that it has no real solution.

A polynomial with zero real roots cannot be factored into linear factors ${\displaystyle x-a}$ where a is a real number. The polynomial ${\displaystyle f\left(x\right)=x^4+6x^3+11x^2+6x+10}$, shown below, has no real roots and therefore cannot be factored using only real numbers.
We need to focus on the word with the highest degree in order to more clearly comprehend the final behavior. Take a fourth-degree polynomial like for example
${\displaystyle f\left(x\right)=x^4+10x^3-x^2+1}$
As x becomes a very large positive number or very large negative number, the ${\displaystyle x^4}$ term will become larger much faster than the other terms, and the other terms will become irrelevant. As ${\displaystyle x^4}$ can never be a negative number if x is real,  On both sides of x, f(x) will get closer to positive infinity. In a similar vein, for a function like
${\displaystyle f\left(x\right)=-x^4-10x^3+x^2-1}$
then ${\displaystyle -x^4}$ will become a very large negative number, regardless of the other terms.

We already mentioned that odd-degree exponents can be either positive or negative in odd-degree functions. Consider, for instance, the function
${\displaystyle f\left(x\right)=x^3+2x^2+3x-1}$
the $x^3$ term will become a very large positive number as x approaches positive infinity, and a very large negative number as ${\displaystyle x\to -\mathrm{\infty }}$, eclipsing the other terms of lower degrees. Similarly, the function
${\displaystyle f\left(x\right)=-x^3-2x^2-3x+1}$
will show the opposite behavior ${\displaystyle -x^3}$ will become a very large negative number as x becomes very large to the positive side, and vice versa. Thus the end behavior of an odd-degree polynomial will always have opposite ends - either it will "start" at ${\displaystyle -\mathrm{\infty }}$ and "end" at ${\displaystyle \mathrm{\infty }}$, or with these directions switched if the leading coefficient is negative.