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.
Explain how to factor the following trinomials forms: x² + bx + c
For x2+bx+c, notice that (x - r1)(x - r2) = x2 - (r1+r2)x + r1r2.
Thus we want integers r1 and r2 with a sum -b and product c.
For example, x2 + 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 = b1 + b2 where the following proportion holds:
a:b1 = b2:c
or equivalently, b1b2 = ac
For example, 2x2 + 11x + 12.
We want b1 + b2 = 11 and b1b2 = 2*12 = 24
We can take b1 = 3 and b2 = 8 and get
2x2 + 11x + 12 = 2x2 + 3x + 8x + 12 = x(2x + 3) + 4(2x + 3) = (x + 4)(2x + 3)
There is formula "ax2 + bx + c = a(x-x1)(x-x2) , x1 and x2 are roots of equation ax2 + bx + c = 0 "
x1,2 = [-b ± √(b2 - 4ac)] / 2a
2x2 + 5x + 3 = 0 , x1,2 = [-5 ± √(25 - 24)] / 4 ,
x1 = (-5 + 1)/4 = -1
x2 = (-5 - 1)/4 = -6/4 = -3/2
2(x-(-1))(x-(-3/2)) = (x+1)(2x+3)
When you have ax2 + bx + c, the quickest way to factor it is to first multiply a by c, and then find two factors that will multiply to equal a*c , add up to b. Then you write (ax + _)(ax + _) with the two factors in the blank spots, then divide it all by a.
Eg. 5x2 + 8x + 3
Here, a*c = 5*3 = 15 and b = 8
Factors that multiply to 15 and add up to 8, are 5 and 3, since 5+3 = 8.
Then write it like so, and you can pull a factor of 5 from the first set of parentheses:
(5x+5)(5x+3) = 5(x+1)(5x+3) but remember we multiplied a by c earlier, so we've actually multiplied the original expression by a (and a = 5), so we need to divide it by a, which we can do to get:
2x² + 5x + 3
a*c = 2*3 = 6 and b = 5
Factors that add up to 5 and multiply to 6, are: 2 and 3.
So write it as:
(2x + 3)(2x+2) and now divide by a (which is 2):
(2x + 3)(x+1)
5x2 - 16x + 3
a*c = 5*3 = 15 and b = -16
the factors are... (-1)*(-15) = 15 and (-1) + (-15) = -16
(5x - 15)(5x - 1) now divide by a (which is 5)
(x - 3)(5x - 1)