Trinomial Factor Forge Worksheet
Algebra 2 & Honors
Trinomial Factor Forge
Unit 2: Polynomial Expressions
Page 1 of 2
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Date:
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Part I: Factoring When \(a = 1\) \(\quad (x^2 + bx + c)\)
Find integers \(p\) and \(q\) such that \(p \cdot q = c\) and \(p + q = b\). Write as \((x + p)(x + q)\).
Check: FOIL expansion
1 \(x^2 + 9x + 20\)
Both Signs Positive
Factor Pairs of \(c = 20\):
Pairs: Sum:
Show Work / Verification:
Factored Form:
2 \(x^2 - 4x - 21\)
Opposite Signs
Factor Pairs of \(c = -21\):
Pairs: Sum:
Show Work / Verification:
Factored Form:
3 \(x^2 - 13xy + 36y^2\)
Honors: Two Variables
Target Coefficients:
\(p \cdot q = 36\) \(p + q = -13\)
Show Work / Note Variable Powers:
Factored Form:
4 \(x^4 - 5x^2 - 36\)
Honors: Quadratic Form (Factor Completely)
Substitution Idea:
Let \(u = x^2\) Factor difference of squares!
Complete Factorization Steps:
Factored Form:
Turn to Page 2 for leading coefficient \(a > 1\) factoring using the \(ac\) grouping method.
Algebra 2 & Honors
Trinomial Factor Forge
Unit 2: Polynomial Expressions
Page 2 of 2
Part II: Factoring When \(a > 1\) \(\quad (ax^2 + bx + c)\)
The \(ac\) Method: 1. Compute product \(a \cdot c\). 2. Find factors summing to \(b\). 3. Split the middle term and factor by grouping.
Rule: Factor out GCF first!
5 \(2x^2 + 11x + 12\)
Standard Grouping
\(ac\) Product & Sum:
\(a \cdot c = 2(12) = 24\) Need sum \(b = 11\) Factors: ___, ___
Split Middle Term & Factor by Grouping:
Factored Form:
6 \(6x^2 - 7x - 5\)
Negative Product
\(ac\) Product & Sum:
\(a \cdot c = 6(-5) = -30\) Need sum \(b = -7\) Factors: ___, ___
Split Middle Term & Factor by Grouping:
Factored Form:
7 \(12x^2 - 28x + 15\)
Higher Composite Values
\(ac\) Product & Sum:
\(a \cdot c = 12(15) = 180\) Need sum \(b = -28\) Factors: ___, ___
Split Middle Term & Factor by Grouping:
Factored Form:
8 \(6x^3 + 21x^2 - 45x\)
Honors: Monomial GCF First
Step 1: Extract GCF
\(\text{GCF} = \text{______}\) Remaining trinomial: \((\quad\quad\quad\quad\quad)\)
Step 2: Factor Remaining Trinomial:
Factored Form:
Always verify answers by expanding factors with FOIL or the distributive property.
Trinomial Factor Forge Answer Key
Teacher Key & Solutions
Trinomial Factor Forge — Solutions
Algebra 2 & Honors
Answer Key: Page 1 of 2
Part I Solutions: Leading Coefficient \(a = 1\)
Standard form \(x^2 + bx + c = (x + p)(x + q)\) where \(pq = c\) and \(p + q = b\).
Grading: 2 pts each (1 pt work, 1 pt answer)
1 \(x^2 + 9x + 20\)
Target: \(c = 20, b = 9\)
Factor Pairs & Sums:
\(1 \cdot 20 \implies \text{Sum} = 21\)
\(2 \cdot 10 \implies \text{Sum} = 12\)
\(4 \cdot 5 = 20 \implies 4 + 5 = 9 \;\checkmark\)
Verification & Teaching Note:
• Check: \((x + 4)(x + 5) = x^2 + 5x + 4x + 20 = x^2 + 9x + 20\)
• Order of binomials does not matter by commutative property of multiplication.
Correct Factored Form:
\((x + 4)(x + 5)\) or \((x + 5)(x + 4)\)
2 \(x^2 - 4x - 21\)
Target: \(c = -21, b = -4\)
Factor Pairs & Sign Analysis:
\(1 \cdot (-21) \implies \text{Sum} = -20\)
\(3 \cdot (-7) = -21 \implies 3 - 7 = -4 \;\checkmark\)
Larger factor must be negative.
Verification & Teaching Note:
• Check: \((x - 7)(x + 3) = x^2 + 3x - 7x - 21 = x^2 - 4x - 21\)
• Common Pitfall: Mixing signs \((x + 7)(x - 3)\) yields \(+4x\), not \(-4x\).
Correct Factored Form:
\((x - 7)(x + 3)\) or \((x + 3)(x - 7)\)
3 \(x^2 - 13xy + 36y^2\)
Honors: Two Variables
Coefficient Strategy:
Need \(p \cdot q = 36\) and \(p + q = -13\)
\((-2) \cdot (-18) = 36 \implies -20\)
\((-4) \cdot (-9) = 36 \implies -13 \;\checkmark\)
Variable Placement & Check:
• Second term requires variable \(y\): \((x - 4y)(x - 9y)\)
• FOIL: \(x^2 - 9xy - 4xy + 36y^2 = x^2 - 13xy + 36y^2 \;\checkmark\)
• Remind students not to omit the trailing \(y\) variable.
Correct Factored Form:
\((x - 4y)(x - 9y)\) or \((x - 9y)(x - 4y)\)
4 \(x^4 - 5x^2 - 36\)
Honors: Factor Completely
Step 1: Quadratic Form
Let \(u = x^2 \implies u^2 - 5u - 36\)
Pairs of \(-36\): \(-9 \cdot 4 = -36\), sum \(= -5\)
\((u - 9)(u + 4) = (x^2 - 9)(x^2 + 4)\)
Step 2: Difference of Squares!
• Notice \((x^2 - 9)\) factors further: \((x - 3)(x + 3)\)
• \((x^2 + 4)\) is a sum of squares (irreducible over \(\mathbb{R}\)).
• Stop warning: Stopping at \((x^2 - 9)(x^2 + 4)\) is incomplete!
