explain factoring trinomials

Question

explain how 3x^2+13x-10 equals (3x+2)(x-5)

 

Answer 1

To factor ax² + bx + c, whenever it is possible, you want to find two integers m and n whose sum is b and product is ac. Then you can readily factor.

Let's try this on your quadratic, 3x² + 13x - 10.

We want two numbers whose sum is 13 and product is -30.

-30 factors in a number of ways: -30*1, -15*2, -10*3, -6*5, -5*6, -3*10, -2*15, and -1*30.

The desired factorization is -2*15 since -2+15 = 13.

Thus,

3x² + 13x - 10 = 3x² - 2x + 15x - 10 = x(3x-2) + 5(3x-2) = (x+5)(3x-2)

Your (3x+2)(x-5) is actually a factorization of 3x² - 13x - 10, but is obtainable the same way.

 

Answer 2

Maybe the best way to prove this is just to multiply (3x+2) times (x-5). We would use the FOIL method, which stands for First, Outer, Inner, Last. Here are the steps:

1. multiply the first, meaning the 3x times the x. This gives us 3x^2.

2. Multiply the outer, which means the 3X times the -5. This gives us -15x.

3. Multiply the inner, which means the 2 times the X. This gives us 2X.

4. multiply the last, meaning the 2 times the -5, giving us -10.

Putting it all together, we have:

3x^2 -15X +2X -10, or 3x^2-13X-10.

To do the reverse, you sort of have to use some trial and error. The only way to factor 3x^2 is to do this:

(3X    )(X    )

then you can deal with the -10.  The only way for two things multiplied together to equal -10 would be 1*-10, or -1*10, or 2*-5, or -5 *2. So you experiment till it works, like this:

Let's just try this one and see if it works: (3X -1)(x+10). Now use FOIL, to get 3x^2+30X-X-10, or 3X^2+29X-10, which is clearly wrong.

Next you might try:

(3x+10)(X-1), which gives 3X^2+-3X+10X-10, or 3X^2+7X-10, which is clearly still not the original polynomial.

If you keep experimenting, you'll eventually get to the right answer.

Over time, the more you practice, the better of a "feel" you'll get, and it will get much faster. Hope this helps!