Review both methods and try to identify their main characteristics.
See solution.
Practice makes perfect
Let's review how to factor both kinds of expressions, and then we can contrast the similarities and differences using a table.
Factoring Expressions of the Form x^2+bx+c
Let's factor the expression x^2+2x-24.
x^2+ bx+ c
x^2 + 2x - 24
To factor this expression we need to find the factors of c such that their sum is b. For this case, we need the numbers whose product is 24 and their sum is 2. Since -24<0, the factors should have different signs. Since 2>0, the greatest factor should be positive. Let's check for these requirements for different factors of -24.
Factors of - 24
Sum
- 1,24
-1 +24 = 23 *
-2,12
-2 + 12 = 10 *
- 3,8
- 3 + 8 = 5 *
-4,6
-4 + 6= 2 âś“
With this in mind, we can factor our original expression.
x^2+2x-24 =(x - 4)(x + 6)
Factoring Expressions of the Form ax^2+bx+c When a≠1
For expressions of this form it is recommended to start by looking for the greatest common factor (GCF). This is the common factor with the greatest coefficient and the greatest exponent. Let's consider the case 3x^2 + 6x-72, for instance. In this case the GCF is 3.
3x^2+6x-72 & = 3(x^2) + 3(2x)- 3(24)
3x^2+6x-72 & = 3 (x^2 + 2x - 24)
We can see that the expression inside the parentheses is the same from the previous case. However, to reduce a expression of the more general form ax^2+bx+c to one of the form x^2 +bx+c is not always possible. If we consider 2x^2+11x+12, for instance, the GCF is 1. Then we have to proceed differently.
ax^2+ bx+ c
2 x^2+ 11x+ 12
We need to look for factors of a c such that their sum is b. In this case a c= 2( 12) = 24. Since 24>0, we will need the factors to have the same sign. 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 + 24 = 25 *
2,12
2 + 12 = 14 *
3,8
3 + 8 = 11 âś“
4,6
4 + 6 = 10 *
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 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 the binomial (2x+3) as a common factor. 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).
Summary and Comparison
We will use a table to summarize the process for both cases, and contrast the similarities and the differences more clearly.
x^2+bx+c
ax^2+bx+c
The trinomial does not have a GCF.
We look for the GCF, and if possible reduce it to the case x^2+bx+c.
We look for factors of c such that they add up to b.
We look for factors of ac such that they add up to b.
We use the factors to rewrite the trinomial as a product of binomials.
We use the factors to rewrite the linear term, obtaining 4 terms.
The process is finished.
We will be able to find a common factor for the first two of terms as well for the last two.
We should be able to find a binomial as a common factor. Factoring this binomial will lead to a product of two binomials as required.
