Expanding vs. Factoring Polynomials
Polynomial Long Division
Class Work: Show all Work!!
Re-bubble answer sheets from last
Solving equations and factoring are the two most important
concepts in algebra. There are concepts we will cover later in
which the first step will be to factor a polynomial. If you can’t
factor the polynomial then you won’t even be able to start the
problem, let alone finish it. For example..
Class Notes: Intro. to Factoring
+ 2x – 3)
+ 6x + 9)
Polynomials: Expanding vs. Factoring
Expanding Polynomials: is simply the multiplication (distribution)
of each term by every other term. At this point, we can expand
by which we determine what we multiplied to get the given
quantity. This we do this all the time with numbers.
8 • 7 = 56
Factoring Polynomials: factoring is the process
A common method of factoring
numbers is to completely factor the
number into positive prime factors.
If we completely factor a number into positive prime
factors there will only be one way of doing it.
Only one of the above factorizations of 12 is a complete factorization.
That is the prime factorization of 12.
Factoring polynomials is done in the same manner. We determine all
the terms that were multiplied together to get the polynomial. We
then try to factor each of the terms we found in the first step. This
continues until we simply can’t factor anymore. When we can’t do
any more factoring, the polynomial is completely factored.
For instance, here are just a few of the ways to factor 12.
Here are four examples: x2
- 20x + 100 =
- 16 = x2
+10x + 25 =
2. If you write a polynomial as the product of two or more
polynomials, you have factored a polynomial. For example:
3. A fully factored polynomial cannot be simplified further.
4. Not all polynomials can be factored!! (Using Whole Numbers)
1. Expanding & Factoring are essentially opposite operations.
5. We will start by factoring the 3 special products because of
their simple, easy to memorize factor patterns.
34 – 9x(x + 2)
+ 12x + 35 = (x + 5)(x + 7)
Square of a sum
Square of a difference
Difference of Squares
Find each Product
Dividing a Polynomial by a binomial
Please take complete, easy to
read notes, you will need them.
Part I: Dividing by a Monomial:
1. Take each term in the numerator and divide by the
denominator. The result is now a monomial divided by a
monomial for each term.
Part II: Dividing by a Polynomial: (Long Division)
We will divide x3
- 5x -2 by x-2; x-2 x3
- 5x -2
Step 1: Write both in standard form. Then find the highest degree
terms in both the divisor and the dividend. In this case, that would
be x and x3
Step 3: Multiply by the Divisor, subtract:
Step 2: Divide x3
by x. The result is:
Place above the x2
in the dividend
Step 4: Repeat the Process:
÷ x =
Step 5: Multiply by the Divisor:x- 2
0 Step 6: Last Division
- 5x -2
+ 3x +1
Quick Check. Multiply: (x – 2)(x2
+ 3x +1)=
- 4 =
**Note: If there are missing degrees, fill in with 0𝑥 𝑑𝑒𝑔𝑟𝑒𝑒
Remainder: If there is a remainder, it is shown
last as the remainder over the divisor.
- 4 Solve.