Example 3: Factor a quartic binomial with two variables by taking a common factor

Factor 4x^4y+8x^3y as 4x^3y(x+2).
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Example 3: Factor a quartic binomial with two variables by taking a common factor

Discussion and questions for this video
Because one times something is just that something. Writing 1x is the same as writing x.
LCM is the lowest common multiple. This means that you find the smallest number that can be divided by both. For example, the LCM of 4 and 3 is 12, and the LCM of 3 and 6 is 6.

LCD is the lowest common denominator. This is useful when adding or subtracting fractions, as you need a LCD to add/subtract them. You would find it in the same method that you would find the LCM. for example, the LCD of 1/2 and 1/3 would be 6. You would change the denominator of both fractions to six and then alter the numerator by the same factor as the denominator. So, 1/2 would become 3/6 and 1/3 would become 2/6. They can now be added or subtracted.

GCF is the largest number that both numbers can be divided by. There really isn't a simple way to find it, but you just try to find the largest common factor to both numbers. For example, the GCF of 12 and 18 is 6, and the GCF of 24 and 25 is 1.

hope this helps!
I dont get the difference of the Greatest Common Factor (GCF) and the Least Common Denaminator (LCD) ? is it the same formula to find it ?
No, you are only using GCF to factor these types of polynomials. Look up GCF and LCM on Khan Academy videos and you will see the difference.
I have to do this x4-x2 of course the 4 And 2 are exponents the book said that if possible factor the polynomial completely ?
Just factor out x^2 from each term in the original polynomial, which becomes x^2(x^2-1). Factoring further it becomes x^2(x+1)(x-1). Hope this helps.

Another way to try this one is using substitution, which can be helpful for tricky factoring. An easy way to tackle this problem is to substitute the lowest exponent value of x (in this case x^2) as another variable, such as y. Then, at the very end of the problem, we can put all our y-variables back into x's. So, set x^2 = y. Now the polynomial becomes y^2 - y^1. Factor out a y^1. Now the polynomial becomes: y(y-1). This cannot be factored any further. Now put back the x's, noting that y=x^2. So, the polynomial becomes: x^2(x^2 - 1) This can be factored further: x^2(x+1)(x-1)
I'm trying to understand how to factor polynomials using GCF but I still don't get it. Can you help me understand it?
Andria,
Welcome to Khan Academy.
Try watching the vidoe again. Sometimes it takes a couple times for things to click.
Sal explains it much better than I can.
You are using the Distributive Property. x*(a+b) = xa +xb, but in reverse that is:
xa + xb = a*(a+b).
You want to find the most common factors that make up the x. And this GCF is then placed outside the parenthesis and everything else is left inside the parenthesis.
If you had just numbers such as
2*3*5*5*7 + 2*3*9
you would factor out everything that is common to both. In this case 2*3 and place that outside the parenthesis so you would get
2*3(5*5*7 + 9).
If instead they were letters and numbers such as
x*x*y*5*3 + x*y*y*5*2
You would still find the all of the common factors, in this case x, y and 5 and place them outside the parenthesis
x*y*5(?+?)
and leave everytning else inside
x*y*5(x*3 + y*2)
I hope that helps
Well, in the video, it never says that you should subtract the exponents when you find your GCF, because I was told that you should subtract the exponents. I understand everything else, but that part of the video when he uses the GCF. Does anybody know the answer to my question?
Because 4 is the greatest common factor between the numbers in each term, the two terms share the variable x and you always have to take the smallest exponent so you get x^3, and the two terms also share the variable y (like before you take the smallest exponent) so you get just y. I hope this makes sense.
At 5:09 he says it as if he is subtracting, how does taking a 4 out of an 8 result in a 2? Is he saying we are dividing? Then at 4:57 why is there nothing resulted instead of a 1?
You divide the coefficient to divide (which is what the lesson is, basically), and you subtract exponents to divide, so Sal did want that two, he just misspoke when talking about his operations
Aleta,
Break the numbers down to their prime numbers
55 = 5*11
22 = 2*11
33 = 3*11
So the only common factor in all three numbers is 11
how would you use this method when you have a problem like: 9x^2y^3 + x^2y^2/ 2x^3y^4 when its in that fraction form and you have to simplify using GCF. How can I find a video on here that shows problems like this or where to find out how to solve problems like this one?
some equations can't be factorised by finding a common factor, for that you will have to use the cross method, completing the square or the quadratic equation.
Wait, does factoring out GCF the same as just normal factoring?
Would the answer be the same if the question asked "factor 4x^4 y+8x^3 y"?
Yes, it's the same thing. When people ask you to factor something, you can just assume that they mean factor out the GCF.
2/3 (30 + 1) =

Use the distributive property:

20 + (2/3) = 20 2/3 or 62/3


----or----

1. Complete the operation on the inside of the parenthesis first, and then multiply.

2/3 ( 30 + 1)
2/3 (31) = 20 2/3 or 62/3

Hope this helps! Terrell
Answer: 5(x-10)
Solution:
Think about this way what is their GCF?
If we factor 5 we get 5 and 1. and when we factor 10 we get 2 and 5. So their GCF is 5. Now that we know the GCF, we factor it out to get 5(x-10).
I assume you mean x^2 - 5x - 6 and "what is" is asking for the solution to the equation.

We can solve using the quadratic equation!

[-b ± √(b^2 - 4ac)] / 2a
FYI √ is square root and ± is plus or minus
If you don't know what a, b, and c are, that's okay. a is coefficient of x^2, b is the coefficient of x, and c is the last number. This equation only works with equations like these.

a = 1
b = -5
c = -6

-(-5) = 5

{5 ± [√ (-5^2) - 4(1)(-6)] } ÷ 2(1)

{5 ± [√ 25 + 24]} ÷ 2
{5 ± √49} ÷ 2
{5 ± 7} ÷ 2

5 + 7 = 12
5 - 7 = -2

12 ÷ 2 = 6
-2 ÷ 2 = -1

So your x-intercepts (solutions) are at 6 and -1.

In other words the final factored form is

(x-6)(x+1) < This is the answer

You're probably thinking, "You said the answers were 6 and -1, but you put x-6 and x+1. Yes I did! That's the correct format. When you set x to 6 in the equation, you get zero and same if you put in -1. Hope this helps :)
what if it is a 2 instead of a 4 or higher? would you say 1 x 1 x whatever...
If there were a two in the first polynomial instead of a four (2x^4y instead of 4x^4y), you would factor it out as: 2*x*x*x*x*y.

1*1*x*x*x*x*y would just be x^4y, the ones are redundant.
These videos are great, but I'm just wondering one thing: why did Sal say stuff like X+X+X+X when he could have said X times 4? I mean, I think a good number of us viewers know that 4x=x times 4.
He didn't say x+x+x+x, he said x TIMES x times x times x. Very different.

X + X + X + X is equal to 4X.
X * X * X * X is equal to x^4.
question what do you do when you have a problem with a variable squared a number and variable and then a number by itself?
80+65
distributive property write each sum as a product of the GCF of the two numbers
Austin
Break each number into its primes
80 = 2*2*2*2*5
65 = 5*13
The only common factor is 5 so factor out a five
5(16+13)

I hope that helps make it click for you.
1. Identify the common factors of 80xy and 245x^2
Factors of 80xy are 5 * 2 * 2 * 2 * 2 * x * y
Factors of 245x^2 are 5 * 7 * 7 * x * x

2. Take the common elements out of this list of factors and determine the number to factor out. Here the number is 5x.

3. Determine how many times 5x goes into 80xy and -245x^2 by dividing these numbers by 5x, then write your result in parentheses: 5x(16y - 49x)

5x(16y - 49x) is your answer
Can you please explain how do i factorize 8(4x+5y)2 +12(4x+5y)

the 2 in above expression is square ..
If I have a more complicated expression that becomes 11g(2^2+x^2) how do I simplify that? I was told only when you are subtracting two squares you can use (a+b)(a-b)
At 3;51 my teacher told me something entirely different. i don't understand either onecan you show me a simpler
Do you have a solution for factors of 9 than GCF and factor of 12 than GCF if you do please tell me
Is there anyway I can find a video related to the FOIL method? Please anyone respond.
At 5:13, how do you get 2 if you take a four out of an 8?
Do you have any videos that tell how to do this when you have a polynomial with four terms, our math teacher shows what you're doing when you have two terms, but when there are four or more, she shows something COMPLETELY different. Thanks :)
There are several factoring techniques. Remove a GCF from a polynomial can be done with a polynomial of 2 or more terms. However, not all polynomials have GCF factors. So, other techniques are needed. There is a technique called grouping this is used for a polynomial of 4 terms. Check out video titled: Example: Basic grouping.
Is there a more simple explanation to this video? Im only in 6th grade and have a test
I have different answer for question find GCF for 4x^4y+8x^3y =4xy(x^3+2x^2)
ok, but what do you do when 4x^4-8x^3y? (subtracted)
On one of the mastery challenges, it says to use this same method to rewrite the expression 60+44 as the product of the greatest common factor of 60 and 44 and the sum of the remaining numbers. How do you do that exactly.
60 = 2 * 2 * 3 * 5 and 44 = 2 * 2 * 11 so the GCF = 2 * 2 = 4

So rewrite 60 + 44 as 4*(15 + 11)
when two or more numbers are fractions, im presuming only fractions and zero are considered factors, is that right?
This video is very nice and informative 😀
What is the steps i take is i have to factor a fraction out instead of a whole number? I.E. (1/2y^2 -9/2y -11) I can get as far as 1/2(y^2-9y-*****) I dont know if i factor a 1/2 from the 11 or not.
Of course you factor 1/2 from the 11. Why would you factor it out of the first two terms but not the last one? What is 11 divided by 1/2?
Well you can if the two numbers were 4 and some multiple of 4 but in this case, the numbers involve variables so you have to account for the variables as well as the numbers you are trying to find the greatest common factor.
Question: Pam had two pieces of cloth the same length. She cut the first one into 8 equal parts. She cut the second one into 12 equal parts. How many parts from the second piece equal the same length as 4 parts from the first.
Suppose you have a question like y^3 +y^2 -y, won't the GCF be y^3?
No, it might be more clear to you, if you would write power as multiplication:
y*y*y + y*y - y. If you look at each of those numbers, the smallest number has only one y, so you can factor only y: y(y^2+y+1)
GCF is y.
At the very end of the video he said if you take a 4 out of an 8 you get a 2, am i missing something or was he not using subtraction?
Not subtraction, but division. If you take a factor of 4 out of 8 you get 2, because 2 times 4 equals 8.
What is the fastest way to factor any problem. EX: Polynomials, Trinomials, Binomials Ect.

Thanks, Herobrine
Start by factoring out the Greatest Common Factor, then it depends on the make-up of the problem to decide whether it can be factored or which method to use. Sorry no better answer here.
OK, so I understand how to factor binomials, but what exactly is the point. When would this be needed?
i have a question...when a math prob is like this: 6y^3 (to the power of..) do they mean to put 2 'y's' or three extra?
At 1:49 seconds, Sal goes through the numbers and variables, but what if you have more than one "2" left? and could someone elaborate on how to write out the answer?
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can you use a gcf calculator to find gcf because i found a website with a gcf calculator. This is the website: http://www.mathsisfun.com/greatest-common-factor-tool.html.
You could, but it wouldn't work for the variables like x and y.

For example, if you typed in 5x and 15x, they would have no idea what to do and they wouldn't give you an answer. But it would work for the numbers alone.
The GCF of two numbers x and y is 3. Can you predict the GCF of 4x and 4y? Explain your answer.
For example if I where to have -3x squared -3x+60 and it asked me find the GCF of the expression. Then factor the expression. How would I do this because it is different then simply factoring a quadratic expression
Hi, it is probably just 3 as the GCF: 3( -x^2 -x +20 ). 3 goes into all the coefficients, and we can't take out any x's because 60 doesn't have one.
What's the difference between HCF, GCF, GCD and HCD?
HCF is the same as GCF, and GCD is the same as HCD.
Highest (H) is exactly the same as Greatest (G) but you'll typically see G used more often.
Watching earlier videos helps a lot too. Reading other people's questions and answers on the videos help a lot too.
completely factor the polynomial.

4x^3+12x^2+4x+12
How to use zeros to write a polynomial function
I have to find the lcm and gcf of xy^2 x xy^ -7. How do I do this and what impact does the negative seven have on the answer?
I have watched this video a few times. Every time I think I have it down I get thrown off. Can someone please help me understand how to factor polynomials using GCF?
I'm a bit confused; not about the actual solving, but the meaning of finding the GCF and factoring. When I was younger, I learned that the GCF is basically the biggest term that can go into a set of numbers. For example:

GCF of 8 and 10 is 2

But this video explains it further with more "complications". For example:

GCF of 8 and 10 is 2(4+5)

So could someone tell me which one is the "real" GCF?
Hello, maybe I can help. In your first set, they are just 2 separate numbers, 8 & 10. In the second example you give it is an expression, 8+10, and not just 8 & 10. It may not make sense what the difference is, but remember that we are working with variables whose values we don't know. What is the GCF of 8 and 10? It is 2 as you said already. What is the GCF of 8x+10? It is 2, and the factored expression then becomes 2(4x+5). Another way to think of the difference is that your first example is to teach you how to find the GCF, while the second is the _application_ of the GCF to _factor_ an expression.
you would take out the GCF between 10 and 25. The multiples of 10 are 1,2,5,10 and the multiples of 25 are 1,5,25. The greatest one that has the same multiple in each of the number is 5. So you would factor out the 5 which the equation would 5(2x^2+5). This is the furthest you can go unless you were to factor out by solving a square.
So, I get how to find the GCF of a polynomial, but how do you find the greatest common monomial factor of a polynomial? Is it sort of the same thing? Or is it different?
I am having math trouble with my math homework with GCF can anyone help me it`s 15y^2-240 and I need factor by using GCF please help me.
The GCF is, in other words, is the greatest number that goes into both 15 and 240. In this case the GCF is 15. It goes into both 15 and 240. So
15y^2-240. Factor out 15.
15(y^2-16). This equals to 15(y-4)(y+4).
I am factoring polynomials, and trinomials using the GCF first, and then grouping. I can't find a video for this. Help please. Thanks!
So does that mean that whatever is left once you find what they have in common is always going to be the answer ?
If you were to factor out the GCF that happened to be a term in the expression, would you completely eliminate it or leave it as 1?
I lost you at about 3:45, I don't understand the things after that.... -__-
Grace,
12p^2 when factored to its primes is 2*2*3*p*p
30q^5 when factored to its primes is 2*3*5*q*q*q*q*q
The only common factors in both are 2*3.
So the Greatest Common Factor = 6

I hope that helps make it click for you.
when is the GCF of two or more numbers equal to one of the numbers?
OK, so you could factor out the 4x to the 3rd y using the first step, but then for the second step when Sal shows that you can take the remaining x and the 2 and put them in parenthesis, where did he get the plus sign from exactly? Is it because 2 is a positive factor and not a negative one?
I have to study for a math test tomorrow and I'm struggling with factoring things out. I also have to memorize sixteen different axioms and propertys. Any tips??
How do I find the sum of the numbers 56+64 as the product of their GCF and another sum
What if one polynomial has a 5 in it and the other has a 10, lets say the question is "5ly + 10l*2 y*2. Can I write "5 times l times y and for the other one 5 time 2 times l times l times y times y? Like can I use different numbers?
I know its basic but it helped me to actually see the ones that are cross cancelled which I did in my notes so that when I did the division I could visualize; 4x^3y(1*x*1 + 2*1*1) since the original monomials have the terms adjacent to them so that they are basically; 4*x^4*y + 8*x^3*y. I think the comments below address this, but I wanted to post this for my own edification and in case it may help someone :)