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# Factoring completely with a common factor

We can factor quadratics by first pulling out a common factor so the result looks like a(x+b)(x+c). Created by Sal Khan.

## Want to join the conversation?

• Does a have to be greater than b when a≠b? For example, 8(x+2)(x+4) versus 8(x+4)(x+2).
• I didn't really understand what you were asking, but 8(x+2)(x+4) and 8(x+4)(x+2) are exactly the same thing. Remember that the order in which you multiply two or more things together doesn't change the final product.
a*b*c = b*a*c = c*a*b

Hope this helps.
• To use this method does a have to be greater than 1?
• It doesn't need to be greater than 1, but if it wasn't greater than 1 then it would be pointless. If the only greatest common factor was 1 for the expression, then you can't really factor the expression therefore making the expression prime.
• can someone explain the a+b and a*b thing? I'm lost.
• When you expand n quantities you get n terms for example
(a+b)(a+b) gives exactly 4 terms, 2 of of them are like terms which you add them up. Sal has already mastered factoring so he can easily recognize pattern which is why he says a+b and a*b.
-3x^2+21x-30 = (-3x+6)(x-5)
Find 2 number that gives 90 when multiplied and 21 when added.
15 and 6
-3x^2+15x+6x-30
-3x(x-5)+6(x-5)
(-3x+6)(x-5)
Is this still correct?
When checked it still gives me the first -3x^2+21x-30. But the final answer way different than the one shown in the video
• You are correct, but yours is not the completely factored form of this polynomial. Sal factors out 3 at the very beginning, and you can see this in yours: in -3x+6, 3 is common.

Hope this helps.
• How do I do this with numbers that have no common factors? My maths teacher says it is possible but I do not understand how.
• How do we 'reverse' the process? eg at , how do we apply the distributive property to end up with the original trinomial 4x^2-8x-12? Does the 4 multiply just the first set of brackets or both of them ? i.e. 4(x-3) and 4(x+1) or 4(x-3) and (x+1) remains as it is before expanding?
• To ´reverse´ the process the 4 is multiplied only to the first set of brackets. Once you multiply the 4 to the first set of brackets multiply the remaining 2 brackets together to get back to the original trinomial. Think of 4 as just a factor. For example, if you had 2x3x4, you would multiply from left to right or use the commutative property to multiply whichever numbers seem more compatible. The trinomial in the video is the same except with variables, and variables are just unknown numbers. You could have 4(x-3) first then (4x-12)(x+1) to get 4x^2-8x-12, or 4(x+1) first then (4x+4)(x-3) to get 4x^2-8x-12. You could even multiply (x+1) and (x-3) together first then multiply (x^2-2x-3) by 4 to get the original trinomial 4x^2-8x-12. All of these different combinations of multiplication work due to the commutative property.
• How do I know if my (ax) factor should be factored out positively or negatively? At he goes from positive to negative.
• Think about it. When you're factoring -3, you're essentially dividing all terms by -3 and then multiplying that -3 again. So, divide -3x^2 by -3. You get x^2. Then, divide +21x by -3. You get -7x. Finally, divide -30 by -3. You get 10. That's how you factor it out.

• I understand how to do it and why u factored out the -3.But would 3(-x+5)(-x+2) be correct ?Sorry if it is a dumb question.
• In a way yes. But usually we don't want to work with negative x's. So, that's why we factor out -3 instead of just 3. I think what you have might be counted as an impartial answer. You still need one more step. Which would be factoring out -1.
• Am I wrong for factoring like this? (I am using the example from )
4x^2 - 8x - 12
4(-12) = -48
4+(-12) = -8
4x^2 + 4x | - 12x - 12
4x(x+1) | -12(x+1)
• @Avonfox

Great Question! Well, you can always check if you factored correctly by the "FOIL" method. If you don't know what foil is, it's First, Outer, Inner/Inside, and Last.

So: (x + 1)(4x - 12)

FIRST: Multiply 4x times x, which is 4x²
OUTER: Multiply x times -12, which is -12x
INNER/INSIDE: Multiply 4x times 1, which is 4x
LAST: Multiply 1 times -12, which is -12

So, your problem should look like this:

4x² - 12x + 4x - 12

Lastly, you'll combine the like terms which are -12x + 4, and that comes out to be -8.

4x² -8x - 12✓

Your factoring method is correct. Great Job!

Hope this helps.