Bell Work
Compute using long division. 3587/32
32 3587
What we already know…
The following statements are “equivalent” for a polynomial f and a real
number k:
1. x = k is a solution (or root) of the equation f(x) = 0.
2. k is a zero of the function f
3. k is an x-intercept of the graph of y = f(x).
Today we will add.
4. (x – k) is a factor of f(x).
Division Algorithm
Let f(x) and d(x) be polynomials with the degree of f greater than or
equal to the degree of d, and d(x) ≠ 0.
Then there are unique polynomials q(x) and r(x), called the quotient
and remainder such that
f(x) = d(x) ● q(x) + r(x)
Where either r(x) = 0 or the degree of r is less than the degree of d.
Example
3x + 2 3x3 + 5x2 + 8x + 7
We can write the
answer in polynomial
form or fraction form.
Example – You Try
Use long division to find the quotient and remainder
when 2x 4  x 3  3 is divided by x 2  x  1.
When the factor looks like (x – k)
f(x) = (x – k)●q(x) + r
Remainder Theorem
If a polynomial function f(x) is divided by (x – k), then the remainder is
r = f(k).
Example
Find the remainder when f(x) = 3x2 + 7x – 20 is divided by (x – 2).
Example – You Try
Find the remainder when f(x) = 3x2 + 7x – 20 is divided by (x + 1).
Example – You Try
Find the remainder when f(x) = 3x2 + 7x – 20 is divided by (x + 4).
Factor Theorem
A polynomial function f(x) has a factor (x – k) if and only f(k) = 0.
Synthetic Division
• A shortcut we can use instead of Long Division.
Example
Divide 2x3 – 3x2 – 5x – 12 by x – 3 using synthetic division.
Example – You Try
Divide 3x2 + 7x – 20 by x + 4 using Synthetic Division.
Rational Zeros Theorem
Zeros of a function are either rational or irrational. The rational zeros
theorem allows us to make a list of all the possible rational zeros for a
polynomial function with integer coefficients.
Suppose f is a polynomial function.
If p/q is a rational zero of f, then
p is an integer factor of the constant coefficient, and
q is an integer factor of the leading coefficient.
Example
• List the possible rational zeros of f(x) = 3x3 + 4x2 – 5x – 2.
• Then check using synthetic division.
Example – You Try
• List the possible rational zeros of f(x) = x3 – 3x2 + 1.
• Then check each one by plugging it in to the function.
Homework
2.4 (pg. 205) #4,8,10,14,16,22,24