Simple explanations of factoring trinomials?

Question

I would like to understand this concept fully please, so if it could be explained thoroughly but also simply, that would be great. Here are three problems I have for examples, and I would appreciate not only the steps being laid out in front of me, but also what you did to reach that next step. So here they are!

#1: x^2-4x-5

#2: 2x^2+5x-3

#3: 6x^2-17x+12

THANK YOU!!!!!!

Answer 1

You should say 2nd degree trinomials, else you could have something like x^17 + x^5 +2 which would be difficult to factor.

Let a = the coefficint of x^2 and b be the coefficient of x and c = the constant term. Then for your 1st problem a = 1, b=-4 and c = -5.

So compute D = b^2-4ac = 16 -(4)(1)(-5) = 16+20 = 36

Since this is a perfect square, we know that we will have two real factors. The roots will be (-b +/- sqrt(D)/(2a) = (4 +/- 6)/2 = -1 or 5

Hence the factors are (x+1)(x-5). This method always works.

If D = 0, you will have two equal factors and if D < 0 no real factors.

For your 2nd problem, you know that the factors (if they exist) will be:

(2x -3)(x+1) or (2x-1)(x+3) because 2x and x are the only factors of 2x^2 and 1 and -3 are the only factors of -3. So look at each one and you will see that only (2x-1)(2x+3) gives the correct middle term 5x.

The 3rd problem gets a little harder because 6 = 6*1 or 2*3 and 12= 12*1 or 6*2 or 4*3. You also note that the middle term is negative. This means that 12 = -2*-3 or -1*-12or -3*-4. So you can play around with the possible combinations until you find the right, or you can use the method shown for problem 1.

Answer 2

When factoring you must try to get the numbers to equal out. So on #1 you need two numbers that multiply together to get -5.  However the same two numbers must add up to -4.  So you would have to use  

-5 and 1 for the second part.  You need x2 for the first part so you have x and x.  That comes out to (x-5)(x + 1)  because -5 + 1 = -4

Apply the same principal  to the others.  #2 you need 2x and x, etc

Answer 3

1) x² - 4x - 5

First you need to find out what items multiplied together are possible:

for x², it's x and x. that's it.

For -5, it's -1, 5 and 1, -5.

So we really have two choices.

Now we do a FOIL check to see which one makes -4.  Here, -5 and 1 does, so your factor is:

(x - 5)(x + 1)

For #2, you have a few more choices:

for 2x²:  2x and x, or x and 2x

for -3:  1 and -3, or -1 and 3

Now you have four possibilities (becuase you have 2 choices of each, 2*2 = 4).  You need to use FOIL to find what makes +5.

Here, (2x - 1)(x + 3) meakes +5 (2x*3 - x = 6x - x = 5x)

The same steps for #3, just more possible choices ..   I'll leave that for you to work out.  Don't forget that to get a +12 that adds to a -17, you'll need to use two negative numbers, since you cannot add positive numbers together to get a negative sum.