Lesson 3.1 Factoring Polynomials
by Susan Johnsey
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FACTORING is a major portion of the math classes required after Algebra 2.
I will begin each of these lessons with easy problems from Algebra 1. But by the end of each lesson you will be working problems that require the same concepts, but that are at an Algebra 2 level.
Be sure you read and complete all examples in these lessons. WRITE them into your notes. You will need them for other Lessons in this course, its sequel and in future algebra or precalculus or calculus courses.

Be sure to write all examples in your notebook.
This chapter grows and depends greatly on prior knowledge.
In simple every day terms factoring polynomials is finding a way to
rewrite the problem as Multiplications.
SEVERAL skills are needed to be able to do this.
There are several "types" of factoring because of all the different types of polynomials.
ARE YOU READY? We will start with GREATEST COMMON FACTORING.
If given 3x 12 can you tell me the monomial and binomial that were multiplied
together to obtain 3x 12?
They were 3 and (x  4). To check this recall Lesson 2.2.
3 times (x  4) =3x  12, that is the distributive property.
What about 5abx +50x ? Think then look below.
5x times (ab + 10).
Hope you see the 5x times the ab is 5abx and also see that the 5x times the +10 is +50x.
So 5abx +50x = 5x(ab+10).
Try these: 5x^{2}y  6xy and also 10w^{2} 5w^{3}+15w.
Find the answers below by the $$$, then return here to continue.
Finding the monomial can be ambiguous because there may be more than one.
So for that reason we will always try to find the monomial
with the largest absolute numeric value
and the largest absolute numeric exponents. That is a mouthful.
What that means, in other words, is that we will find the greatest common factor of all the terms.
Looking back at 5x^{2}y  6xy, a monomial we could have used was
x times the binomial 5xy 6y
or the monomial y times the binomial 5x^{2} 6x. Both of the pairs multiply back to give the 5x^{2}y  6xy.
But we must choose to use xy , that is the greatest common factor,
( x and y are just common factors).
Remember we will always use the greatest common factor of all the terms.

For 10w^{2} 5w^{3}+15w we use 5w, the greatest common factor.
5 alone or w alone would be just common factors.
We must use the GREATEST!

$$$ Answers for the above practice problems are: xy times 5x  6
and 5w times (2w  w^{2}+ 3).

Here are the steps to always follow first when
finding Greatest Common Factor.
Example: 42w^{2}x  6wx +36x .
1. There are three terms.
2. Look at the three coefficients: 42 and 6 and +36.
They all have in common what number factor? 6
3. NOW look at the three terms.
They all have in common what variable factor? x
and What is the lowest value of the exponent of x? 1
4. So the greatest common factor for all three terms is 6x.
Factor out the 6x from the three terms:
Divide (opposite of multiply) all the terms by the 6x to
find the 3 new terms inside the parenthesis.
Please look closely at the exponents and variables. Recall x divided by x = 1.
42w^{2}x  6wx +36x Divide each term by the 6x.
Divide the 42w^{2}x BY 6x
and the  6wx BY 6x
and the +36x BY 6x.
We get 7w^{2} w + 6 .
Answer is 42w^{2}x  6wx +36x = 6x(7w^{2} w + 6)
Multiply the three terms inside the parenthesis by the 6x to check your answer.

Example : The greatest common factor is 7x^{2}y
for 14x^{4}y  42x^{3}y 63x^{2}y .
CAN YOU show that 14x^{4}y  42x^{3}y63x^{2}y = 7x^{2}y( 2x^{2} 6x 9) ?
The steps for finding the 7x^{2}y :
1. There are 3 terms to consider.
2. They all have in common what number factor? 7
3. They all have in common what variable factor? x and also y
and What is the lowest value of the exponent of x? 2
and What is the lowest value of the exponent of y? 1
4. So the greatest common factor for all three terms is 7x^{2}y.
Factor out the 7x^{2}y from the three terms.
Divide (opposite of multiply) all the terms by the 7x^{2}y to find the 3 new terms inside the parenthesis.
Please look closely at the exponents and variables.
Hope you can see the colors here.
Divide the 14x^{4}y by 7x^{2}y
AND the  42x^{3}y by 7x^{2}y
AND the 63x^{2}y by 7x^{2}y = 2x^{2} 6x  9
That is
14x^{4}y  42x^{3}y  63x^{2}y = 7x^{2}y ( 2x^{2} 6x 9)
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Example using 1:
The greatest common factor is 5x for 5x + 35yx^{3}.
5x + 35yx^{3} = 5x(1 + 7yx^{2}) ,
Do not forget the 1.
Do you see that the 5x times 1 gives us the original 5x?
And the 5x times the 7yx^{2} gives us the +35yx^{3}^{ } ?
Write these in your notes!!
READ CAREFULLY. ALGEBRA 2 problems coming below!
Be sure you understand what is going on with the exponents before going to the next example; that is, when do we add them and when do we subtract them, and when do we multiply them.
Do you know what x x equals?
What about x^{2a} x^{a } ?
Each of these involve multiplying,
there is a common base x ,
and each has an exponent.
Do you know how to simplify x^{3}x^{2 }? ^{ }
is this x^{5}or x^{6 . } Where are your notes from Lesson 2.2?
LOOK:^{ }x^{3}x^{2 }is (x x x) (x x)
x^{3}x^{2 = }x^{5}
We are multiplying so we can add the exponents!! = x^{5 .}
BUT NOW LOOK:
How do you simplify^{ }(x^{3}) ^{2 }?
the^{ 2} means write the x^{3}down twice. (x^{3}) ^{2 =}(x^{3})(x^{3})^{ }
We now have a multiplying problem so we add the exponents =x^{6} ^{.}
Or you can recall:
when an exponent is "on an exponent" then
we can multiply the exponents.
(x^{3})^{2} = x^{6}

Now back to my original questions:
Do you know what x x equals? That is the same as x^{1}x^{1} .
Of course it is x^{2}. We ADDED the exponents.
What about x^{2a} x^{a } ?
This is multiplying so what do we do with the exponents? we add!!
xx = x^{2} and x^{2a}x^{a} = x^{3a} Know these backwards and forwards!!!
Who can get this one right?
**Does x^{5a}= x^{5}x^{a} or does x^{5a}= x^{3a}x ^{2a} ?
Add the exponents to find out.
** which one above equals 5a?
IS it 5+a or is it 3a+2a?
I hope you know 5+a is adding UNLIKE terms
and we can NOT add these unlike terms.
Answer: x^{5a}= x^{3a}x^{2a} but also we know: x^{5a}= x^{4a}x^{a}.

Recall when we multiply terms we add the exponents.
Thus when we factor (the undoing of multiplying) we subtract the exponents of the
GCF from the exponents of the terms.
We must of course be subtracting the "corresponding" exponents.
A common factor for X^{3a} X^{a+1}+ 4X^{a}^{ }is X^{a}. Use the steps from above.
1. There are 3 terms to consider.
2. They all have in common what number factor? only a 1
3. They all have in common what variable factor? X
and What is t he lowest valueof the exponent of X?
We will say 1a, but that really is tricky since we do not know its value.
Most algebra books will use the 1a.
4. So the greatest common factor for all three terms is 1X^{a}.
Factor out the 1X^{a }from the three terms.
Divide the three terms by X^{a}. (that is subtract the a from the exponents)
X^{3a} X^{a+1} + 4X^{a} =
X^{a } ( X^{3a} ^{ a}  X^{a+1}^{ a}^{ } + 4X^{a}^{ a} ) =
HOPE you see that 3a  a = 2a NOT 3. !!!
The 3a  a is same as 3a  1a which = 2a.
X^{a}(X^{2a}X^{1}+4X^{0}) = Recall x^{0} is 1.
X^{a}(X^{2a} X^{1}+ 4 ).
To check, multiply the 3 terms inside the parenthesis by X^{a}.
To multiply we add the exponents.

Do Assignment 3.1 .
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