Download Algebra 2, with Trig

Advanced Functions
3.1 Remainder and Factor Theorems
Name______________________ Date __________
We’ve learned how to multiply polynomials. Now, we’ll learn how to DIVIDE polynomials.
There are 2 ways to divide: long division and synthetic division. First, let’s go back to grade school and review
how to divide plain old regular numbers.
8 65180
Now, let’s apply the same process to polynomials.
x - 2 x  2x  6x  9
3
Try
2
2x + 3 2 x  7 x  17 x  3
3
2
Check your solution by multiplying the divisor
by the quotient and adding the remainder.
Synthetic division works differently, but is pretty cool!
3
2
Divide x  2 x  6 x  9 by x – 2
1. First, the divisor must be in the form: (x – k) So, in our example, k = 2. What is k in the divisor (x + 4)?
2. You only need the coefficients of the terms. However, if a term is missing, you must use a zero as a place
holder.
3. Write the leading coefficient. Then multiply and add, multiply and add, etc.
2
1
2
2
-6
8
-9
4
1
4
2
-5
4. The answer is interpreted as follows:
Work backwards.
The last number is the remainder.
The next number back is the constant.
The next number back is the x coefficient.
2
The next number back is the x coefficient.
3
The next number back is the x coefficient.
And so on.
So, the answer above is
2
1
1
x
Try it.
2
2
2
4
-6
8
2
-9
4
-5
x
c
remainder
Divide x
3
2
 x + 4x + 2 +
14 x  8 by x + 4
5
x2
Remainder Theorem:
In the example on the previous page, divide x  2 x  6 x  9 by x – 2, we got x + 4x + 2 +
3
2
2
5
x2
The Remainder Theorem states that f(2) will be equal to –5!
Try it.
Factor Theorem:
The factor theorem just uses definitions you already know!
Solutions = Roots = Zeros = X-Intercepts
When we find the roots of a polynomial equation, we are finding the places where the value of the function is
ZERO. If f(x) = 0, then x is a root!!!!
Now we have to connect this idea with synthetic substitution. Remember that synthetic substitution allows us to
find the value of a function for a given value of x.
Watch what happens when we find f(3) for the function f(x) = x2 + 2x − 15
3
1
2 −15
3 15
1 5
0
The value of the function is 0!! That means 3 is a root of
the equation, a zero of the function, and an x-intercept on the
graph!! That also means that (x − 3) is a factor of the polynomial. And it means
that (x + 5) is also a factor, also know as a reduced polynomial.
Wow!! Getting zero in synthetic substitution is a big deal!!
Study this example: f(x) = x3 + x2 + 2x + 24.
Graph the function. (You will need to set your window.) There is one real root at −3. Because the degree of the
polynomial is 3, we know there are two other roots. They must be imaginary.
We will use synthetic substitution to divide out (x + 3). Then the quadratic formula will allow us to find the
other roots.
−3
1
1 2
−3 6
1 −2 8
24
−24
0
So, using the coefficients of the quotient, we write
(x + 3)(x2 − 2x + 8) = 0
We already know x = − 3. We use the quadratic formula
2  4  4(1)(8)
2   28
on the second ( ) and get
=
=
2
2
2  2i 7
= 1  i 7 . So, x = −3, 1  i 7 , 1  i 7
2
This stuff is not hard. It just takes practice to remember the process!
(AKA – Do your HW!)