Factoring Trinomials of the Form ax 2 + bx + c, where a  Factor ... - Free Printable

Let’s look at what’s happening in each example. We’re multiplying two binomials (like (3x - 2) and (x + 4)) to get a trinomial (like 3x² + 10x - 8). The question is: “What still works?” — meaning, what pattern or rule stays true when we factor trinomials of the form ax² + bx + c?

Looking at the examples:

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First example:
(3x - 2)(x + 4)
= 3x·x + 3x·4 -2·x -2·4
= 3x² + 12x - 2x - 8
= 3x² + 10x - 8

Notice:
- First term: 3x² → comes from 3x · x
- Last term: -8 → comes from -2 · 4
- Middle term: +10x → comes from 12x - 2x

Also note: signs are different in the factors (-2 and +4), and the last term in the trinomial is negative (-8).

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Second example:
(5x + 2)(x - 3)
= 5x·x + 5x·(-3) + 2·x + 2·(-3)
= 5x² -15x + 2x -6
= 5x² -13x -6

Again:
- First term: 5x² → 5x · x
- Last term: -6 → 2 · (-3)
- Middle term: -13x → -15x + 2x

Signs in factors are different (+2 and -3), and last term in trinomial is negative.

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Third example:
(4x + 3)(x + 5)
= 4x·x + 4x·5 + 3·x + 3·5
= 4x² + 20x + 3x + 15
= 4x² + 23x + 15

Here:
- First term: 4x² → 4x · x
- Last term: +15 → 3 · 5
- Middle term: +23x → 20x + 3x

Signs in factors are the same (both positive), and last term in trinomial is positive.

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So what “still works”? Let’s think about factoring backwards.

When you have a trinomial like 3x² + 10x - 8, you want to find two binomials that multiply to give it.

The key patterns that still work:

✔ First term (ax²): Comes from multiplying the first terms of the binomials. So if you have 3x², possible first terms could be 3x and x.

✔ Last term (c): Comes from multiplying the last terms of the binomials. So if c = -8, possible pairs are:
- (+1, -8), (-1, +8), (+2, -4), (-2, +4)

✔ Middle term (bx): Comes from adding the “outer” and “inner” products. That’s why we check which pair of factors for ‘c’ gives us the right middle term when combined with the first terms.

✔ Sign rules:
- If the last term (c) is negative, then the signs in the binomials are different.
- If the last term (c) is positive, then the signs in the binomials are the same (and they match the sign of the middle term b).

These rules still work even when ‘a’ is not 1 (like 3x², 5x², etc.). You just have more possibilities to test for the first terms.

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Final Answer:
The rules that still work when factoring trinomials of the form ax² + bx + c (even when a ≠ 1) are:
1. The first term (ax²) comes from multiplying the first terms of the binomials.
2. The last term (c) comes from multiplying the last terms of the binomials.
3. The middle term (bx) comes from adding the outer and inner products.
4. Sign rules:
- If c is negative → binomial signs are different.
- If c is positive → binomial signs are the same (and match the sign of b).