Factoring Fixers Worksheet
Factoring Forensics
Official Lab Report: GCF Cases
Agent Name: ___________
Date: ___________
MISSION BRIEF: Below are 5 "solved" GCF cases. However, the previous agent made critical errors. Identify the mistake, provide the correct factorization, and explain the error to prevent future mishaps.
CASE #01: \(6x^2 - 12x\) Status: Compromised
Suspect's Work:
\(6(x^2 - 2x)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #02: \(10y^3 + 15y^2\) Status: Compromised
Suspect's Work:
\(5y(2y^2 + 3y)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #03: \(8a^4 - 4a^2\) Status: Compromised
Suspect's Work:
\(4a^2(2a^2)\)
Investigator's Correction:
Error Identification & Explanation:
Factoring Forensics
Official Lab Report: Mixed Cases
CASE #04: \(14b + 21\) Status: Compromised
Suspect's Work:
\(7(2b + 21)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #05: \(9m^2n - 18mn^2\) Status: Compromised
Suspect's Work:
\(9mn(m - 9n)\)
Investigator's Correction:
Error Identification & Explanation:
Phase II: Trinomial Investigations (\(a=1\))
CASE #06: \(x^2 + 5x + 6\) Status: Compromised
Suspect's Work:
\((x + 6)(x - 1)\)
Investigator's Correction:
Error Identification & Explanation:
Factoring Forensics
Official Lab Report: Trinomial Cases
CASE #07: \(x^2 - 7x + 10\) Status: Compromised
Suspect's Work:
\((x - 5)(x + 2)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #08: \(x^2 + 3x - 10\) Status: Compromised
Suspect's Work:
\((x + 2)(x - 5)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #09: \(x^2 - 10x + 25\) Status: Compromised
Suspect's Work:
\((x - 5)(x + 5)\)
Investigator's Correction:
Error Identification & Explanation:
CASE #10: \(x^2 + x - 12\) Status: Compromised
Suspect's Work:
\((x + 3)(x - 4)\)
Investigator's Correction:
Error Identification & Explanation:
End of Lab Report — File #882-FX
Factoring Fixers Slides
Top Secret
Factoring
Forensics
Case File: GCF & Trinomial Investigations
Authorized Personnel Only
Mission Briefing
Current Situation
Several factoring "solutions" have been processed incorrectly. Our math lab is compromised.
Your Objective
Examine the "Suspect's Work," identify the procedural error, and provide a verified correction.
10
Cases to Close
Evidence Type A: GCF
1
Find Greatest Factors
Look for the largest number and variable power that divide all terms.
2
Extract the GCF
Place the GCF outside the parentheses.
3
Divide & Conquer
Divide every original term by the GCF to find the remaining expression.
\(12x^3 - 8x^2 \rightarrow 4x^2(3x - 2)\)
Criminal Profile: The Partial GCF
Critical Error
The Evidence:
\(15y^4 + 20y^2\)
Suspect's Solution:
\(5(3y^4 + 4y^2)\)
What happened?
The suspect found the greatest numeric factor (5), but ignored the variables.
Corrective Action:
\(5y^2(3y^2 + 4)\)
Evidence Type B: Trinomials (\(a=1\))
M
Multiply to 'c'
Find two numbers whose product is the constant term.
A
Add to 'b'
The SAME two numbers must sum to the middle coefficient.
Target Profile:
\(x^2 + bx + c\)
\((x + \text{?})(x + \text{?})\)
Criminal Profile: The Sign Saboteur
Critical Error
The Evidence:
\(x^2 - 2x - 15\)
Suspect's Solution:
\((x + 5)(x - 3)\)
The Fatal Flaw
The suspect found factors that multiply to -15, but they sum to +2. The target was -2.
Corrective Action:
\((x - 5)(x + 3)\)
Go To Work
Open your Lab Reports. Analyze all 10 cases.
Accuracy is mandatory. Speed is secondary.
SESSION STARTING IN... 3... 2... 1...
Factoring Fixers Answer Key
Confidential Case Summary
Teacher's Guide & Answer Key: Factoring Forensics
INTERNAL USE ONLY
Analysis Overview
This document contains the verified solutions for the 10 factoring cases. Use this to provide immediate feedback to agents (students) or to facilitate a debrief on common misconceptions in factoring.
Phase I: GCF Cases
Case
Polynomial
Correct Result
Error Analysis (The "Crime")
01
\(6x^2 - 12x\)
\(6x(x - 2)\)
Partial GCF. The suspect failed to factor out the variable term (x).
02
\(10y^3 + 15y^2\)
\(5y^2(2y + 3)\)
Smallest Exponent Rule. The suspect only pulled out \(y\) instead of \(y^2\).
03
\(8a^4 - 4a^2\)
\(4a^2(2a^2 - 1)\)
Placeholder Malfunction. The suspect "deleted" the second term instead of leaving a 1.
04
\(14b + 21\)
\(7(2b + 3)\)
Division Failure. The suspect found the GCF but didn't actually divide the 21 by 7.
05
\(9m^2n - 18mn^2\)
\(9mn(m - 2n)\)
Factor Confusion. The suspect used factors of 18 (2 and 9) rather than dividing 18 by 9.
Phase II: Trinomial Cases
Case
Polynomial
Correct Result
Error Analysis (The "Crime")
06
\(x^2 + 5x + 6\)
\((x + 2)(x + 3)\)
Product Error. The suspect picked factors that sum to 5 (6, -1) but multiply to -6.
07
\(x^2 - 7x + 10\)
\((x - 5)(x - 2)\)
Mixed Sign Error. To multiply to +10 and sum to -7, both factors must be negative.
08
\(x^2 + 3x - 10\)
\((x + 5)(x - 2)\)
Sign Swap. The suspect used (+2, -5) which sums to -3 instead of +3.
09
\(x^2 - 10x + 25\)
\((x - 5)^2\)
Difference of Squares Confusion. The suspect used conjugates (+5, -5) resulting in middle sum 0.
10
\(x^2 + x - 12\)
\((x + 4)(x - 3)\)
Larger Number Sign. The positive sum means the factor with the larger absolute value must be (+).
Investigator Field Notes
The "Check Your Work" Strategy
Encourage agents to "re-distribute" or FOIL their factored answer. If it doesn't match the original case evidence exactly, the criminal is still at large.
