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 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)