how do you understand factoring

Question

Struggling a little in school and want to really do well this last semester.

Answer 1

Here's an example problem. Factor this formula:

x²+3x-10

First I'll give you the solution, then we'll look into how to get there.

A quadratic formula that can be factored will give you the product of two binomials:

(x+a)(x+b)

This one factors to:

(x-2)(x+5)

 

If we do this in reverse, we can see how to factor the quadratic.

Multiply the two binomials using FOIL.

(x-2)(x+5) = X²+5X-2X-10        = x²+3x-10

 

The number parts of the binomials are (+5) and (-2).

Notice that the -10 at the end is the product of the (-2) and the (+5).

Those same two numbers are coefficients of the two x-terms in the middle: (+5x) and (-2x).

When you add these two together, they become 5x-2x = 3x.

That 3 in the 3x is the sum of the (+5) and the (-2).

 

Now, if we think about this a little, we can see what has to happen to factor a quadratic.

Look at the last term in the quadratic.  It's -10.

We need to find two numbers that equal -10 when multiplied together.

Those same two numbers have to add up to equal the coefficient of the x-term.  That's +3 in this case.

Since the -10 is negative, one of its factors will be positive and one negative.

 

x²+3x-10

We need two factors of -10 that add up to +3.

Some possible factors are
-1 and 10   these add up to 9.  No help.

2 and -5     these add up to -3, but we need +3.

-2 and 5     these add up to +3.   That's it!!

(x-2)(x+5)  We factored it!

 

Remember.  If the last term is negative, you get two factors with opposite signs, one positive and one negative.

If it's positive, both factors will be positive or both negative.

 

I hope this helps.

Gene