When factorizing a expression of the form ax^2+bx+c we need two numbers whose product is ac. How can we decide which possibility will be the correct choice?
See solution.
Practice makes perfect
Recall that to factor a expression in the form ax^2+bx+c we start by looking for two numbers such that their product is equal to ac. To pick the correct possibility we need to consider which of these options has a sum that can give b. If they follow that condition the polynomial will be able to be factored by grouping. Let's see an example.
ax^2+bx+c
2 x^2+11x+12
We need to look for factors of ac such that their sum is b. In this case ac=(2)(12) = 24. Since 24>0 we will need the factors to have the same sign. And, since 11>0 we know both should be positive. Let's check for these requirements for different factors of 24.
Factors of 24
Sum
1,24
1 + 12 = 11 *
2,12
2 + 6 = 4 *
3,8
3 + 8 = 11 ✓
4,6
3 + 4 = *
Now we rewrite our original expression using the factors that met the requirements mentioned before.
2x^2+11x +12 = 2x^2+3x+8x +12
We continue by looking for the greatest common factor, or GCF, of the first two terms and the two last terms.
2x^2+3x+8x +12 = x(2x+3)+4(2x+3)
Finally, notice that both expressions have as a common factor the binomial(2x+3). We can factor the expression once more.
x(2x+3)+4(2x+3) = (2x+3)(x+4)
We found that the expression 2x^2+11x+12 can be factored as (2x+3)(x+4).
