Explain how

to factor the following trinomials forms: x² + bx + c (Example: x² + 4x + 4) and ax² + bx + c (Example: 2x² + 5x + 3). Be specific and show your answer using both

words and mathematical notation.

For x^{2}+bx+c, notice that (x - r_{1})(x - r_{2}) = x^{2} - (r_{1}+r_{2})x + r_{1}r_{2}.

Thus we want integers r_{1} and r_{2} with a sum -b and product c.

For example, x^{2} + 7x + 10.

Notice that the two integers with a sum -7 and a product of 10 are -2 and -5

Thus we get (x+2)(x+5)

For the more general case ax2+bx+c, if it factors then we can write b = b_{1} + b_{2} where the following proportion holds:

a:b_{1} = b_{2}:c

or equivalently, b_{1}b_{2} = ac

For example, 2x^{2 }+ 11x + 12.

We want b_{1} + b_{2} = 11 and b_{1}b_{2} = 2*12 = 24

We can take b_{1} = 3 and b_{2} = 8 and get

2x^{2} + 11x + 12 = 2x^{2} + 3x + 8x + 12 = x(2x + 3) + 4(2x + 3) = (x + 4)(2x + 3)

## Comments

You can divide either set of parentheses by

a(that's where the 2 came from, see the other examples where i divide bya), but for convenience, divide the one that'll give you whole numbers, not fractions. We divide byabecause we multiplied byawhen findinga*c.And yes, this method works for all trinomials.

"if I divide (2x + 3)(2x+2) by 2 that is totally different than just dividing the second set of equations."

It won't matter where you divide, as long as you divide (2x + 3)(2x+2) by 2. You could divide the first parentheses by 2 to get:

(x + 1.5)(2x+2) which is equal to (2x + 3)(x+1) - try FOILing them both.

This is just one of the properties of division. If you have (a*b)/2, it is equal to (a/2)*b and a(b/2).