"We will be examining quadratic trinomials of the form ax2 + bx + c."

bullet Part 2: Trinomials with a ≠ 1 (ax2 + bx + c)

 


Unfortunately, when the leading coefficient is a number other than one (and there is no common factor among the terms), the number of possible answers increases, making the search for the correct answer more difficult.

Method by Trial and Error:
Factor: 3x2 - 7x - 6
Step 1: The first step when factoring is to always check for a common factor among the terms. In this example, there is no common factor.
Nope!
Step 2: Consider all of the possible factors of the leading coefficient. In this example, there is only one way to arrive at 3x2: the first terms of the factors will need to be 3x and x.
(3x        )(x        )
Step 3: The product of the last terms must be -6. Unfortunately, there are several ways to arrive at this product of -6.
 
+6 and -1
-6 and +1
+3 and - 2
-3 and +2
ALL of these pairings
yield a product of -6.

Notice in this step how it was necessary to consider each of these pairings with BOTH arrangements of 3x and x. Be sure you cover ALL of the possibilities.
Possible answers?
(3x + 6)(x - 1)
(3x - 6)(x + 1)
(3x + 3)(x - 2)
(3x - 3)(x + 2)
(x + 6)(3x - 1)
(x - 6)(3x + 1)
(x + 3)(3x - 2)
(x - 3)(3x + 2)
Step 4: Only one of the possible answers shown above will be the correct factoring for this problem. Which choice, when multiplied, will create the correct "middle term" needed for this example?
(The needed middle term for this problem is "-7x".)
(3x + 6)(x - 1) middle term +3x
(3x - 6)(x + 1) middle term -3x
(3x + 3)(x - 2) middle term -3x
(3x - 3)(x + 2) middle term +3x
(x + 6)(3x - 1) middle term +17x
(x - 6)(3x + 1) middle term -17x
(x + 3)(3x - 2) middle term +7x
(x - 3)(3x + 2) middle term -7x
Step 5: Write the answer.
3x2 - 7x - 6 = (x - 3)(3x + 2)
This trial and error method may take some time to accomplish. Once you determine the possible factors for the leading term and for the constant term,
concentrate on finding the correct middle term. Be careful of the signs!

Unfortunately, the "pattern method" we saw with trinomials with a leading coefficient of a = 1, does not apply to situations where a ≠ 1!

shortcutnopic
While there is really no quick "short cut method" for factoring
ax2 + bx + c when a ≠ 1, you do have some options that you may find helpful. Check out factoring using the "ac" method.


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Here is a bit of good news!!

It may be the case, in a few situations, that a trinomial with a leading coefficient other than "1", may actually have a leading coefficient of "1", after you factor out the Greatest Common Factor!

Method: Leading Coefficient of "1" in Hiding!!
Factor: 2x2 + 4x - 48
Step 1: The first step when factoring is to always check for a common factor among the terms. This trinomial contains a greatest common factor of 2.
Factor out 2 from each term.
2(x2 + 2x - 24)
Step 2: YEA!!! We now have a trinomial with a leading coefficient of "1" and we can apply our Pattern Method".
We are looking for numbers that multiply to -24 and add to +2.
The numbers must be +6 and -4.
Don't forget to carry along the 2, we factored out at the beginning, into your final answer.
2(x + 6)(x - 4)

DONE!

 

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The graphing calculator can prove to be helpful for checking your answer when factoring over the set of integers.
ti84c

For help with factoring on your calculator,
Click Here!

 

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