How to factor a trinomial by getting rid of the impostor

Learn how to factor a trinomial of the form ax2 + bx + c when a is not equal to 1 by getting rid of the "impostor". GCF(a,b,c) = 1 

Factoring a trinomial by getting rid of the impostor
The impostor here is a, or the coefficient of ax2 . The impostor will show itself in the factored form as

a(     )(      )     or      b(      )c(      )     if a = bc

This method is straightforward and does not involve too much trial and error and that is why I like it. The best way to understand what we mean by getting rid of the impostor or a is to solve some more examples. Here are 3 more examples showing how to factor a trinomial quickly.

More examples showing how to factor a trinomial

Example 1: 3x2 + 7x + 2

For this example, the impostor that we will eventually get rid of is 3.

Step 1

Factor any GCF if any. Since 3x2 + 7x + 2 does not have any GCF, go to step 2.

Step 2

Multiply a by c.

a × c = 3 × 2 = 6

Step 3

Factor as if a = 1 using (3x + blank)(3x + blank) as your factored form. Then, ask what multiplies to 6 and adds up to 7.

Notice how we used 3x as the first term in each factor! This is the way to do it every time. 

Since 1 × 6 = 6 and 1 + 6 = 7, use 1 and 6 to fill in the blank.

We get (3x + 1)(3x + 6)

Step 4

Pull out impostor by GCF and banish. In other words, using (3x + 1)(3x + 6), factor out the GCF for each factor.

(3x + 1)(3x + 6) = (3x + 1)×3(x + 2)

As you can see, the impostor is the 3, so banish it.

After banishing or getting rid of the impostor which is 3, the final answer is (3x + 1)(x + 2)

Example 2: 8x2 - 10x - 3

For this example, the impostor that we will eventually get rid of is 8.

Step 1

Factor any GCF if any. Since 8x2 -10x -3 does not have any GCF, go to step 2.

Step 2

Multiply a by c.

a × c = 8 × -3 = -24

Step 3

Factor as if a = 1 using (8x + blank)(8x + blank) as your factored form. Then, ask what multiplies to -24 and adds up to -10.

Notice again how we used 8x as the first term in each factor!

Since -12 × 2 = -24 and -12 + 2 = -10, use -12 and 2 to fill in the blank.

We get (8x + -12)(8x + 2)

Step 4

Pull out impostor by GCF and banish. In other words, using (8x + -12)(8x + 2), factor out the GCF for each factor.

(8x + -12)(8x + 2) = 4(2x + -3)×2(4x + 1)

As you can see, the impostor is split up into two of its factors, so banish it by multiplying the two numbers to get your real impostor, 8.

After banishing or getting rid of the impostor which is 8, the final answer is (2x + -3)(4x + 1)

How to factor a trinomial when there is a GCF in step 1


Example 3: 6x2 + 8x + 2

Step 1

Factor the GCF if any. GCF = 2, so 6x2 + 8x + 2 = 2(3x2 + 4x + 1 )

Step 2

Be careful here. The impostor is 3 since we need to factor 3x2 + 4x + 1.

Multiply a by c.

a × c = 3 × 1 = 3

Step 3

Factor as if a = 1 using (3x + blank)(3x + blank) as your factored form. Then, ask what multiplies to 3 and adds up to 4.

Since 3 × 1 = 3 and 3 + 1 = 4, use 3 and 1 to fill in the blank.

We get (3x + 1)(3x + 3)

Step 4

Pull out impostor by GCF and banish. In other words, using (3x + 1)(3x + 3), factor out the GCF for each factor.

(3x + 1)(3x + 3) = (3x + 1)×3(x + 1)

After banishing the impostor, we get (3x + 1)(x + 1)

However, do not forget the factor in step 1, so the final answer is 2(3x + 1)(x + 1)

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