Factor Fortress Worksheet
Factor Fortress
Breach the walls of complexity using the power of factoring.
Knight:
Date:
The Challenge
To capture the fortress, you must factor each expression completely. Show your work clearly to avoid the traps of calculation errors.
I The Outer Walls (Trinomials where \(a=1\))
Problem 1
\(x^2 - 14x + 45\)
Problem 2
\(x^2 + 3x - 108\)
Problem 3
\(x^2 - 25x + 150\)
Problem 4
\(x^2 + 19x + 84\)
II The Inner Gate (Trinomials where \(a > 1\))
Problem 5
\(6x^2 + 7x - 20\)
Problem 6
\(12x^2 + 5x - 2\)
Problem 7
\(15x^2 - 19x + 6\)
Problem 8
\(8x^2 - 22x + 15\)
III The Great Keep (Advanced \(a > 1\))
Problem 9
\(10x^2 + 31x + 15\)
Problem 10
\(24x^2 - 10x - 21\)
IV The Secret Vault (Factoring by Grouping)
Problem 11
\(15x^3 - 10x^2 + 12x - 8\)
Problem 12
\(21x^3 + 6x^2 - 28x - 8\)
Problem 13
\(18x^3 - 27x^2 + 8x - 12\)
Problem 14
\(24x^3 - 16x^2 - 15x + 10\)
Problem 15
\(42x^3 + 48x^2 + 7x + 8\)
Factor Fortress Answer Key
Factor Fortress
Master Key • Teacher Resource
ANSWER KEY
Section I: Trinomials (\(a=1\))
1. \(x^2 - 14x + 45\) \((x - 9)(x - 5)\)
2. \(x^2 + 3x - 108\) \((x + 12)(x - 9)\)
3. \(x^2 - 25x + 150\) \((x - 15)(x - 10)\)
4. \(x^2 + 19x + 84\) \((x + 7)(x + 12)\)
Section II: Trinomials (\(a > 1\))
5. \(6x^2 + 7x - 20\) \((3x - 4)(2x + 5)\)
6. \(12x^2 + 5x - 2\) \((4x - 1)(3x + 2)\)
7. \(15x^2 - 19x + 6\) \((5x - 3)(3x - 2)\)
8. \(8x^2 - 22x + 15\) \((4x - 5)(2x - 3)\)
Section III: Advanced (\(a > 1\))
9. \(10x^2 + 31x + 15\) \((5x + 3)(2x + 5)\)
10. \(24x^2 - 10x - 21\) \((6x - 7)(4x + 3)\)
Section IV: Grouping
11. \(15x^3 - 10x^2 + 12x - 8\) \((5x^2 + 4)(3x - 2)\)
12. \(21x^3 + 6x^2 - 28x - 8\) \((3x^2 - 4)(7x + 2)\)
13. \(18x^3 - 27x^2 + 8x - 12\) \((9x^2 + 4)(2x - 3)\)
14. \(24x^3 - 16x^2 - 15x + 10\) \((8x^2 - 5)(3x - 2)\)
15. \(42x^3 + 48x^2 + 7x + 8\) \((6x^2 + 1)(7x + 8)\)
Factoring Tactics Slides
FACTOR FORTRESS
Siege Tactics for Complex Quadratics
\(a=1\) Tactics
\(a>1\) Strategies
Grouping Manoeuvres
The Outer Wall: \(a = 1\)
When factoring \(x^2 + bx + c\), we seek two integers that satisfy two conditions simultaneously:
-
- Their product equals \(c\)
-
- Their sum equals \(b\)
Tactical Example
\(x^2 - 14x + 45\)
Product = 45 (-9) \(\times\) (-5)
Sum = -14 (-9) + (-5)
Result: \((x - 9)(x - 5)\)
The Inner Gate: \(a > 1\)
The AC Method
When the leading coefficient isn't 1, we use a more sophisticated siege engine.
Step 1: Multiply \(a \times c\).
Step 2: Find factors of \(ac\) that sum to \(b\).
Step 3: Split the middle term and factor by grouping.
\(6x^2 + 7x - 20\)
1. \(ac = 6 \times (-20) = -120\)
2. Factors of -120 that sum to 7? 15 and -8
3. Split: \(6x^2 + 15x - 8x - 20\)
4. Group: \(3x(2x + 5) - 4(2x + 5)\)
Final: \((3x - 4)(2x + 5)\)
The Secret Vault: Grouping
Factoring by grouping is used when there are four terms.
The Code
"Group the first pair, group the second pair. Find the common factors, and find the common binomial."
Watch for the negative signs in the second group!
\(15x^3 - 10x^2 + 12x - 8\)
1
Group: \((15x^3 - 10x^2) + (12x - 8)\)
2
GCF: \(5x^2(3x - 2) + 4(3x - 2)\)
3
Binomial: \((5x^2 + 4)(3x - 2)\)
The Fortress Awaits
You have the tactics. You have the engines. Now, breach the walls of complexity and claim your victory.
Knight's Checklist
Check for GCF first
Signs are critical
Distribute to check
Keep work organized
