Factoring General Trinomials

Factoring General Trinomials: A Step-by-Step Guide

1. Introduction (Objective)

Have you ever tried solving a quadratic equation and gotten stuck because it didn’t factor neatly or seemed unfactorable? Or wondered how breaking down trinomials helps real-life problem-solving, like optimizing areas or calculating trajectories?

Hi, I'm Mr. Del.. In this blog article, you’ll learn:

• How to factor general trinomials of the form ax² + bx + c

• A foolproof method to avoid common mistakes

• Real-world applications of factoring

2. Prerequisite Knowledge Check

Before diving in, make sure you’re familiar with:

• Multiplying binomials (FOIL method)

• Factoring simple trinomials (x² + bx + c)

• Greatest Common Factor (GCF)

Need a refresher? Check out these lessons:

• Factoring Basics

• FOIL Method Explained

3. Core Concept Explanation

What is a General Trinomial?

A general trinomial is a polynomial with three terms, typically expressed in the form:

$$\mathrm{<span\; style="background-color:\; \#fcff01;"><b>a</b></span>}{\mathrm{<span\; style="background-color:\; \#fcff01;"><b>x</b></span>}}^{\mathrm{<span\; style="background-color:\; \#fcff01;"><b>2</b></span>}}<span\; style="background-color:\; \#fcff01;"><b>+</b></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><b>b</b></span>}\mathrm{<span\; style="background-color:\; \#fcff01;"><b>x</b></span>}<span\; style="background-color:\; \#fcff01;"><b>+</b></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><b>c</b></span>}$$

where:

• $a$, $b$, and $c$ are coefficients (constants), with $a\mathrm{\ne }0$,

• $x$ is the variable.

This is the standard form of a quadratic trinomial. Trinomials can also appear in higher degrees, such as:

$$\mathrm{<b>a</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>n</b>}}<b>+</b>\mathrm{<b>b</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>m</b>}}<b>+</b>\mathrm{<b>c</b>}$$

where $n>m$ are positive integers. However, the quadratic trinomial ($a{x}^2+bx+c$) is the most common.

A general trinomial has the form: ax² + bx + c (where a≠1)

Factoring a Quadratic Trinomial

A quadratic trinomial $\mathrm{<span\; style="background-color:\; \#fcff01;">a</span>}{\mathrm{<span\; style="background-color:\; \#fcff01;">x</span>}}^{\mathrm{<span\; style="background-color:\; \#fcff01;">2</span>}}<span\; style="background-color:\; \#fcff01;">+</span>\mathrm{<span\; style="background-color:\; \#fcff01;">b</span>}\mathrm{<span\; style="background-color:\; \#fcff01;">x</span>}<span\; style="background-color:\; \#fcff01;">+</span>\mathrm{<span\; style="background-color:\; \#fcff01;">c</span>}$ can often be factored into two binomials:

$$(dx+e)(fx+g)$$

such that:

• $d\times f=a$
  

• $e\times g=c$

• $dg+ef=b$

Example:

Factor $2x^2+7x+\mathrm{3\; =\; (2x\; +\; 1)(x\; +\; 3)}$

$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$$

Factoring Method: The AC Split Method

1. Multiply a and c → Find two numbers that multiply to a × c and add to b.

2. Split the middle term using these numbers.

3. Factor by grouping.

Common Mistakes to Avoid!!!

• Forgetting to check for a GCF first.

• Incorrectly splitting the middle term.

• Misplacing signs (+/-).

4. Worked Examples (Step-by-Step)

Example 1: Easy (a =1)

Factor: x² + 5x + 6

1. Find two numbers that multiply to 6 and add to 5 → 2 & 3.

2. Write as: (x + 2)(x + 3)

Example 2: Medium (a>1)

Factor: 2x² + 7x + 3

1. a × c = 2 × 3 = 6

2. Find two numbers that multiply to 6 and add to 7 → 6 & 1.

3. Split: 2x² + 6x + x + 3

4. Group: (2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)

5. Final factored form: (2x + 1)(x + 3)

Example 3: Hard (Negative Terms)

Factor: 3x² – 10x – 8

1. a × c = 3 × (–8) = –24

2. Find two numbers that multiply to –24 and add to –10 → –12 & 2.

3. Split: 3x² – 12x + 2x – 8

4. Group: (3x² – 12x) + (2x – 8) = 3x(x – 4) + 2(x – 4)

5. Final factored form: (3x + 2)(x – 4)

5. Practice Problems (With Solutions)

Test your skills! Try factoring these:

1. x² + 8x + 12

2. 3x² + 10x + 7

3. 4x² – 12x + 9

Solutions:

1. (x + 2)(x + 6)

2. (3x + 7)(x + 1)

3. (2x – 3)(2x – 3) or (2x – 3)²

6. Real-World Applications

Factoring trinomials helps in:

• Engineering – Calculating forces and trajectories

• Finance – Modeling profit/loss equations

• Architecture – Optimizing area and dimensions

7. Summary & Key Takeaways

• Always check for a GCF first.

• Use the AC Split Method for ax²+bx+c.

• Double-check signs when splitting terms.

• Practice makes perfect—keep factoring!

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Below are the solved problems with detailed solutions.

Example 1: x² + 5x + 6

Step 1: Find two numbers that multiply to 6 (c) and add to 5 (b).
            → 2×3 = 6 and 2 + 3 = 5

Step 2: Write as binomials: (x + 2)(x + 3)

Final Answer: (x + 2)(x + 3)

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Example 2: x² + 7x + 10

Step 1: Find numbers for 10 (product) and 7 (sum).
            → 5×2 = 10 and 5 + 2 = 7

Step 2: Write binomials:(x + 5)(x + 2)

Final Answer: (x + 5)(x + 2)

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Example 3: x²- 4x + 4

Step 1: Find numbers for 4 (product) and -4 (sum).
          → (-2)(-2) = 4 and (-2) + (-2) = - 4

Step 2: Write as perfect square: (x - 2)(x - 2) or (x - 2)²

Final Answer: (x - 2)²

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Example 4: x² + x - 12

Step 1: Find numbers for -12 (product) and 1 (sum).
          → 4×-3 = -12 and 4 + (-3) =1

Step 2: Write binomials: (x + 4)(x - 3)

Final Answer: (x + 4)(x - 3)

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Example 5: x² - 9x + 18

Step 1: Find numbers for 18 (product) and -9 (sum).
            → -6 ×-3 = 18 and -6 + (-3) = -9

Step 2: Write binomials: (x - 6)(x - 3)

Final Answer: (x - 6)(x - 3)

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Additional Example with Partial Solutions. Please fill in the missing parts in the blank.

Example 1: x² + 9x + 20

Step 1: Find two numbers that multiply to __ and add to __.
         → __ × __= 20 and __ + __ = 9

Step 2: Fill in the binomials: (x + __ )(x + __ )

Final Answer: ________

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Example 2: x² - 11x + 24

Step 1: Find numbers for product __ and sum __.
           → __ × __ = 24 and __ + __ = -11

Step 2: Complete the factoring: (x - __ )(x - __ )

Final Answer: ________

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Example 3: x² + 2x - 15

Step 1: Identify numbers where product = __  and sum =__.
            → __ × __ = -15 and __ + __ = 2

Step 2: Fill in the signs and numbers: (x __ )(x __ )

Final Answer: ________

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Example 4: x² - 16

Special Case: Difference of squares (x² - a²). Do a bit of research on this, please.

Step 1: Recognize pattern: x² - ( __ )²

Step 2: Apply formula (x +__ )(x - __ )

Final Answer: ________

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Example 5: 2x² + 7x + 3

Step 1: Multiply a×c: __ × __ = __

Step 2: Find numbers: __ × __ = 6 and __ + __ = 7

Step 3: Split middle term: 2x² +__ x +__ x +3

Step 4: Factor by grouping: ( ____ )( ____ )

Final Answer: ________

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Further Reading & Resources

• Khan Academy: Factoring Trinomials

• Algebra Essentials Workbook

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