Section: 3.3 / Dividing Polynomials: Remainder and Factor Theorems / Day 1 of 1
Think About It A moth has moved into your bedroom closet. She appeared in your bedroom at night, but somehow her relatively stout body escaped your clutches. Within a few weeks, swarms of moths have invaded your now-tattered wardrobe which suggest that Mama Moth was in the family way. You estimate that there must be 200 of the little flying critters nestled into every crevice of your clothing. Two hundred moth-tykes from one female moth – is it possible?
Indeed it is. The number of eggs, f (x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by .
How could we figure out how the width of Mama Moth’s abdominal area?
Long Division of Polynomials and the Division Algorithm
Long Division of Polynomials
Divide by
Long Division of Polynomials
Divide by .
Express the form: quotient plus remainder divided by divisor.
Long Division of Polynomials
Divide by
Dividing Polynomials Using Synthetic Division
Let’s revisit Exampe 2 above and try to divide another way. Divide by
Note how the final quotient (with any remainder) is determined.
For a full detailed explanation of this process, you can consult your textbook – page 346.
Using Synthetic Division
Use synthetic division to divideby .
The Remainder Theorem
If we consider the division algorithm when is divided by , we see the following result:
Try evaluating the function f at c.
Using the Remainder Theorem to Evaluate a Polynomial Function
Given , use the Remainder Theorem to find
The Factor Theorem
Let’s look again at the division algorithm when is divided by .
By the Remainder Theorem, the remainder, r, is , so we can substitute for r.
Try this.
Notice that if , then
This proves that is a factor of . So, if you ever have a remainder, , that is zero, will be a factor of .
Using the Factor Theorem
Solve the equationgiven that is a zero of
Think About It REVISITED
Recall that the number of eggs, f (x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by .
You got bored. So you decided to count all those little moth-tykes. You take pride in that your estimate of 22 was close – there are actually 211 baby moths chomping away at your clothes.
What was the width of Mama Moth’s abdominal area?