College Algebra – 3.2 & 3.3 – Synthetic Division and Zeros of Polynomial Functions
First we will review long division:Divide:
We are going to be dividing A LOT so you can see that it would be very time consuming if we had to long division over and over. That’s why we need a faster way – synthetic division. Synthetic division is just a way to do the division without having to “worry” about the variables. The only drawback is that synthetic division only works if your divisor is linear.
Divide using synthetic division:(3.2, #8)
Now write the answer using the division algorithm. That is, in the form .
If you understood the last problem you have the basic idea of section 3.2. Sometimes the problems will just be worded differently to try to get you ready for the next section.
Example: Express in the form where . (3.2, #22)
Note that this is the same as dividing by ______.
Now we will use the Remainder Theorem to find the value of a function.
Example: Use the remainder theorem to find where . (3.2, #34)
A number is a zero of a polynomial if and only if . In the above example is 4 a zero of f ?
Example: Is a zero of ?
Now write in the form . Do you see that you have factored out ? Now factor the polynomial completely(as a product of linear factors).
Section 3.3
In the previous example we actually worked a problem that will be found in section 3.3.
Factor Theorem
The polynomial is a factor of the polynomial if and only if .
The following statements are equivalent:
k is a zero of.
is a factor of .
is an x-intercept of the graph of .
Now the question becomes, “Is it always possible to factor a polynomial?” The answer is, “YES!” Actually, this is what the Fundamental Theorem of Algebratells us. In fact, every polynomial can be written as a product of linear factors. The problem is that we will not always know a zero ahead of time to help us get it factored. That is why we need the Rational Zeros Theorem.
Rational Zeros Theorem
If is a rational number written in lowest terms, and if is a zero of f , a polynomial function with integer coefficients, then p is a factor of the constant term and q is a factor of the leading coefficient.
Example: Factor and list all zeros: (3.3, #40)
Sometimes you will get complex zeros.
Factor and list all zeros:
Note that the complex zeros occur in conjugate pairs.
Now we will go backwards. Meaning that we will be given the zeros and we need to find the original polynomial.
Example: Find a polynomial function of smallest degree that has zeros 5, -3, and such that .