Question

How do you factor `3x^2+36x+105`?

Answer 1

`3(x+5)(x+7)`

Explanation:

`"take out a "color(blue)"common factor "3`

`=3(x^2+12x+35)`

`"to factor the quadratic"`

`"the factors of + 35 which sum to + 12 are + 5 and + 7"`

`=3(x+5)(x+7)`

Answer 2

`3(x+5)(x+7)`

Explanation:

A quadratic polynomial can be factored if it has zeroes: you can write

`ax^2+bx+c = a(x-x_1)(x-x_2)`

if `x_1` and `x_2` are solutions of the polynomial. So, let's see if our polynomial has solutions: the quadratic formula yields

`x_{1,2} = \frac{-36\pm\sqrt(1296 - 1260)}{6} = \frac{-36 \pm 6}{6}`

So,

`x_1 = \frac{-36+6}{6} = -5`

`x_2 = \frac{-36-6}{6} = -7`

Thus, `3x^2+36x+105 = 3(x+5)(x+7)`