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The first step to factor a trinomial of the form ax^2 +bx +c is to look for factors of ac that add up to b.
See solution.
We will first explain how to find two trinomials with the requested form, and then we will show how to factor them.
Notice that the trinomial given is of the form ax^2 +bx +c. ax^2+ bx+ c 1x^2+ â–ˇx+ 24 The first step to factor a trinomial of this form is to look for factors of a c = 24 that add up to b. We can use a table to organize and check the different possibilities.
| Factors of 24 | Sum of Factors |
|---|---|
| 24,1 | 25 |
| 12,2 | 14 |
| 6,4 | 10 |
| 8,3 | 11 |
The sum of each of these pairs is a valid possibility to write a factorable trinomial of the form needed. Two examples are shown below. 1. x^2+14x+24 2. x^2+11x+24
Now, let's work out how to factor each of the examples given.
As we can see from the table above, 12* 2 =24 and 2 + 12 = 14. Then, we can rewrite 14x as 12x+2x and make groups of two factors with the terms to factor by grouping. x^2+14x+24 ⇕ x^2+2x+12x+24 ⇕ (x^2+2x)+(12x+24) Let's find the greatest common factor (GCF) of each group.
| Rewrite terms as a product of their prime factors. | GCF |
|---|---|
| x^2 &= x * x 2x &= 2 * x | x |
| 12x &= 2 * 2 * 3 * x 24 &= 2 * 2 * 2 * 3 | 2 * 2 * 3= 12 |
Now, we can factor out the corresponding GCF from each group. (x^2+2x)+(12x+24) ⇕ x (x+2)+12 (x+2) As we can see, there is a common binomial factor (x+2). We can factor this binomial out to complete the factoring process. x(x+2)+12(x+2) ⇕ (x+2)(x+12)
As we can see from the table above, 8* 3 =24 and 3 + 8 = 11. Then, we can rewrite 11x as 8x+3x and make groups of two factors with the terms to factor by grouping. x^2+11x+24 ⇕ x^2+3x+8x+24 ⇕ (x^2+3x)+(8x+24) Let's find the GCF of each group.
| Rewrite terms as a product of their prime factors. | GCF |
|---|---|
| x^2 &= x * x 3x &= 3 * x | x |
| 8x &= 2 * 2 * 2 * x 24 &= 2 * 2 * 2 * 3 | 2 * 2 * 2= 8 |
Now, we can factor out the corresponding GCF from each group. (x^2+3x)+(8x+24) ⇕ x (x+3)+8 (x+3) As we can see, there is a common binomial factor (x+3). We can factor this binomial out to complete the factoring process. x(x+3)+8(x+3) ⇕ (x+3)(x+8)