Review how to factor a quadratic trinomial ax^2+bx+c, where a≠1.
Example Solution: 3x^2-10x+8=(3x-4)(x-2)
Practice makes perfect
We are asked to write a quadratic trinomial that can be factored.
ax^2+bx+ c
The coefficients of our trinomial should meet three conditions: a≠1, a c>0, and b<0. We will first review how to factor a quadratic trinomial with a≠1. Then, we will write our trinomial and factor it.
Reviewing How to Factor a Trinomial With a≠1
When factoring a trinomial with a≠1 there are three steps to follow.
Find a c.
Find factors of a c such that their sum is equal to b.
Using the factors found in Step 2 rewrite the linear term bx as two terms.
If we are able to find factors of a c whose sum is equal to b, we know that ax^2+bx+ c can be factored.
Writing the Trinomial
Recall that, by the given requirements, a c has to be greater than zero. Therefore, its factors must have the same sign — both positive or both negative. In particular, a and c are factors of a c, so they must have the same sign.
a>0 and c>0
or
a<0 and c<0
Let's arbitrarily choose the values of a and c.
a= 3 and c= 8
The value of a c is 3* 8=24. As we mentioned earlier, factors of a c must have the same sign. Since b has to be less than zero, those factors will need to be negative so that their sum is negative.
Factors of 24
Sum of Factors
- 1 and - 24
- 1+(- 24)=- 25
- 2 and - 12
- 2+(- 12)=- 14
- 3 and - 8
- 3+(- 8)=- 11
- 4 and - 6
- 4+(- 6)=- 10
Any number from the right column of the above table can be used as b. Let's choose b=- 10. Finally we can write our trinomial by substituting a= 3, b=- 10, and c= 8 into the general formula.
3x^2+(- 10)x+ 8 ⇔ 3x^2-10x+8
By how this trinomial was constructed, we know that it can be factored.
Factoring the Trinomial
We know that a c=24 and that the factors of 24 whose sum is b=- 10 are - 4 and - 6. Knowing this, we can write bx as two terms.
3x^2-10x+8 ⇔ 3x^2 - 4x - 6x+8
Finally, we will factor the last expression obtained.
