Question 484414: Hello, I'm having trouble trying to factor each polynomial completley. The equation is 6x^3+4x^2-10x
In words that equation would be six x cubed plus four x squared minus ten x
Thank you so much for the help.
-Kaitlin
Found 2 solutions by tinbar, ewatrrr:
Answer by tinbar(133)   (Show Source):
You can put this solution on YOUR website! In the last solution I gave you, we dealt with the form of a^2-b^2, which is quick and easy to factor.
Now this poly is a little more general and requires a bit more thinking to solve, but once you understand the concept, you can use to factor all kinds of crazy polynomials.
First, we always try to factor out 'obvious' things. In 6x^3+4x^2-10x, two things are obvious: 1) Every term has a factor of 'x', so we can definitely remove that, and 2) Every term has an even coefficient, meaning we can pull out AT LEAST a factor of 2 [in other cases it's possible to pull out more, we are simply trying to take out the greatest common factor between all the terms].
So now 6x^3+4x^2-10x becomes 2x(3x^2 + 2x - 5). However, this is not completely factored yet, (3x^2 + 2x - 5) can be (possibly) factored more, let's try.
(3x^2 + 2x - 5) = (ax+b)*(cx+d), where a,b,c,d are numbers (they can be negative as well, the + signs are there just as default, they might becomes - signs). Now we have to figure out what a,b,c, and d are. We already have certain clues, for instance, a*c MUST equal 3 since when we re-expand these two brackets a*c gives us the leading term and we already know it must be 3. Similarly, we can see b*d MUST equal -5 since when we re-expand b*d gives us our last term. Now if you have been checking along as I've asked you to, you should be able to take (ax+b)*(cx+d), and expand it and be able to conclude that the middle term coefficient +2, should be given by a*d+b*c.
So we have:
a*c=3
b*d=-5
a*d+b*c = 2
We will just guess since there are only few choices to satisfy these conditions, maybe a=3 and c=1, and b=-5 and d = 1, but if this was the answer, then a*d+b*c = 3*1+(-5)*1 = 3-5 = -2, so that's not the answer we are looking for, we want +2. We just have to make a slight adjustment; let a=3,c=1,b=5, and d=-1[we switched the negative sign of b to d, this is ok since we just require b*d have a final product of -5, where the '-' comes from makes no difference. With this solution we see a*d+b*c = 3*(-1)+5*1 = -3+5 = 2.
So our factored form is: 6x^3+4x^2-10x = 2x(3x^2 + 2x - 5) = 2x
Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi,
Factoring: Expressing as the product of two binomials:
6x^3+4x^2-10x
factoring: First noting each term has 2x in common:
2x(3x^2 + 2x - 5)
2x(3x +5)(x-1) |Fully factored
Checking our factoring using FOIL
F First terms 3x^2
O Outside terms -3x
I Inside terms +5x
L Last terms -5
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