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## Algebra I (2018 edition)

### Course: Algebra I (2018 edition)>Unit 15

Lesson 3: Factoring polynomials by taking common factors

# Taking common factor from binomial

Explore the process of factoring polynomials using the greatest common monomial factor. This involves breaking down coefficients and powers of variables to find the largest common factor, and then rewriting the expression with this common factor factored out. It's an essential skill for simplifying and solving algebraic expressions.

## Want to join the conversation?

• what if it was a subtraction problem like 8x^2y-12xy^2 would the answer still be 4xy(2x+3y) or 4xy(2x-3y)?
• Your 2nd is correct. You can verify it by redistributing the 4xy. Your 1st version doesn't recreate the original polynomial.

Note: you could factor out -4xy. If you did, you would then have: -4xy(-2x+3y)

Hope this helps.
• how would you do 6x^2+11x-10
• What two numbers multiply to be 6*-10 = -60 (which tells us one is positive and one is negative) and subtract to be 11? We end up with 15 and - 4, so 6x^2 + 15x - 4x - 10, then get GCF of first two and of second two to get a common factor, try it and see.
• why is factoring important?
• You can use factoring to help solving quadratic or even higher degree equations a lot of times without using the proper formula like (-b+-sqrt(b^2-4ac))/2a saving a lot of time and unnecessary calculation.

For example: 4x^3-2x^2-6x = 0
you can solve:
2x(2x^2-x-3)=0 //factoring 2x
2x(2x^2+2x-3x-3)=0 //using grouping method
2x(2x(x+1)-3(x+1))=0 //factor 2x from the 1st 2 term; 3 from the 2nd 2 term
2x(2x(x+1)-3(x+1))=0 //factoring (x+1)
(2x)(x+1)(2x-3)=0
Applying rule: A product is zero when some of its factor is zero.
Either one of the 3 must be 0.
I. 2x=0 -> x=0
II. x+1=0 -> x=-1
III. 2x-3=0 -> x=3/2

So you just solved a cubic equation without using any higher college level math. Not working always but certainly an useful skill to learn in high school math.
• Can you use this method for polynomials with more than two or three terms?
• Yep. Think of it like distributing in reverse, which is kinda what factoring is. so 12x^6+8x^4-4x^2 = 4x^2(3x^4+2x^2-1). So you can distribute that 4x^2 and see it turns back into the original expression. But more importantly it shows that method works.
• 15a^2+12a^3
how do you work this on out
• what goes into a^2 and a^3? a^2 because a^2 * 1 = a^2 and a^2 * a = a^3, so we can factor out an a^2 so this gets us a^2 (15 + 12a)

Now we deal with the 15 and 12? what number goes into both of them? 3. if you have trouble with that you need to work on your factors. so if we factor out a 3 we divide both by 3. This leaves us with:

3a^2 (5 + 4a)

You can try distributing again to check. Now, since 5 and 4a do not have any common factors, this is as simple as it gets.

You can find factors by writing a number in prime factorization. If that is not something you get let me know and I can explain it.
• How come when you factor two of the binomials like 30k^5+6k^2 the even exponent can just be taken out of the 30k^5 and be written now as 30k^3? Why wouldn't you divide instead of subtract?
• We can't and aren't using subtraction. k^5 - k^2 are unlike terms. So, subtraction is not possible.

It is division!
k^5/k^2 = (k*k*k*k*k) / (k*k)
Cancel / divide out 2 instances of "k". You end up with k^3.
This is a basic property of exponents. When you divide and the factors have a common base, you subtract the exponents.

Hope this helps.
• (4xy)(2x) + (4xy)(3y) = (4xy)(2x+3y), I don't really understand how they are equivalent, I understand how to do it but is there any other way of demonstrating it turning into that without drawing two arrows and saying you factor out the 4xy?
(1 vote)
• First you need to understand the distributive property (you might already know this):
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
For any three quantities 𝑎, 𝑏, and 𝑐. Going from the LHS to the RHS is known as distributing, while the reverse process (going from the RHS to the LHS) is known as factoring. For instance let 𝑎 = 3, 𝑏 = 5, and 𝑐 = 7. This gives us:
3(5 + 7) = 3 • 12 = 36
But we can also use the distributive property to get the correct answer as well:
3(5 + 7) = (3 • 5) + (3 • 7) = 15 + 21 = 36
We get the correct answer both ways thanks to the distributive property.
Now we make the substitutions:
𝑎 = 4𝑥𝑦
𝑏 = 2𝑥
𝑐 = 3𝑦
And we get:
4𝑥𝑦(2𝑥 + 3𝑦) = (4𝑥𝑦)(2𝑥) + (4𝑥𝑦)(3𝑦)
As desired. Comment if you have questions!
• What is the factored form of this problem x2+8x+15?
• Find 2 numbers that add to be 8 and multiply to be 15, then put them in ( x + __)(x + __). 15 only has two choices.