Polynomial Product Worksheet
Algebra I • Polynomial Arithmetic
Polynomial Product Lab
Mastering the Distributive Property & FOIL Pattern
Name:
Date:
Period:
1 Part 1: The Distributive Property (Monomial × Polynomial)
Rule: \(x^a \cdot x^b = x^{a+b}\)
Core Property: \( a(b + c) = a \cdot b + a \cdot c \)
Example: \( 2x(3x - 4) = (2x \cdot 3x) - (2x \cdot 4) = \mathbf{6x^2 - 8x} \)
- Scaffolded
\( 3x(2x + 5) \)
\( = (3x \cdot 2x) + (3x \cdot 5) \)
Final Simplified Answer:
- Watch Signs
\( -4y^2(3y - 7) \)
Final Simplified Answer:
- 3 Terms
\( 2a(5a^2 - 3a + 4) \)
Final Simplified Answer:
2 Part 2: The FOIL Method (Binomial × Binomial)
First • Outer • Inner • Last
F First Terms
\( (ax + b)(cx + d) \rightarrow ax \cdot cx \)
O Outer Terms
\( (ax + b)(cx + d) \rightarrow ax \cdot d \)
I Inner Terms
\( (ax + b)(cx + d) \rightarrow b \cdot cx \)
L Last Terms
\( (ax + b)(cx + d) \rightarrow b \cdot d \)
Key Step: Combine like terms (typically Outer + Inner) to write in standard form: \( Ax^2 + Bx + C \).
4. Multiply: \( (x + 6)(x + 2) \) Guided
F: \( x \cdot x = \)
O: \( x \cdot 2 = \)
I: \( 6 \cdot x = \)
L: \( 6 \cdot 2 = \)
Combine Like Terms:
Standard Form Answer:
5. Multiply: \( (2x + 3)(x - 5) \) Signs
F: \( 2x \cdot x = \)
O: \( 2x \cdot (-5) = \)
I: \( 3 \cdot x = \)
L: \( 3 \cdot (-5) = \)
Combine Like Terms:
Standard Form Answer:
Algebra I: Multiplying Polynomials Page 1 of 2 • Continue on reverse →
POLYNOMIAL PRODUCT LAB | Page 2: Independent Practice & Modeling
Student Name:
3 Part 3: Independent FOIL Practice
Show all intermediate terms before combining
6. Multiply: Binomial Product
\( (3x - 2)(x + 4) \)
Work Area (F + O + I + L):
Standard Form Answer:
7. Multiply: Dual Negatives
\( (4y - 3)(2y - 5) \)
Work Area (F + O + I + L):
Standard Form Answer:
8. Multiply: Special Product
\( (x + 7)(x - 7) \)
Work Area (Notice like terms):
Standard Form Answer:
9. Multiply: Binomial Squared
\( (3a - 4)^2 = (3a - 4)(3a - 4) \)
Work Area (Expand & FOIL):
Standard Form Answer:
4 Part 4: Geometric Area Modeling
Area = Length × Width
10. Community Garden: A rectangular plot has length \( (3x + 4) \) meters and width \( (2x - 1) \) meters.
a) Write a polynomial expression for the area \( A(x) \) in standard form:
b) If \( x = 5 \text{ meters} \), find the actual numerical area in \(\text{m}^2\):
Garden Blueprint
\( 3x + 4 \) \( 2x - 1 \) Area = \( L \cdot W \)
Units in meters
11 Level Up: Binomial × Trinomial Challenge
Multiply: \( (x + 2)(x^2 - 3x + 5) \)
Final Answer:
12 Error Detective Critique
A student claimed \( (x + 5)^2 = x^2 + 25 \). Identify the error and find the true product.
Correct Product:
Polynomial Product Lab • Distributive Property & FOIL Page 2 of 2
Polynomial Product Answer Key
Teacher Resource Answer Key & Rubric
Polynomial Product Lab — Key
Full Step-by-Step Solutions • Common Misconceptions • Grading Guidelines
Total Points: 30 pts Page 1: 14 pts | Page 2: 16 pts
1 Part 1: Distributive Property Solutions (2 pts each • 6 pts total)
Criteria: 1 pt intermediate distribution, 1 pt final simplified form
1. \( 3x(2x + 5) \)
\( = (3x \cdot 2x) + (3x \cdot 5) \)
\( = 6x^{1+1} + 15x \)
Correct Answer:
\( 6x^2 + 15x \)
2. \( -4y^2(3y - 7) \)
\( = (-4y^2 \cdot 3y) - (-4y^2 \cdot 7) \)
Note: \( (-4) \cdot (-7) = +28 \)
Correct Answer:
\( -12y^3 + 28y^2 \)
3. \( 2a(5a^2 - 3a + 4) \)
\( = (2a \cdot 5a^2) + (2a \cdot -3a) + (2a \cdot 4) \)
\( = 10a^3 - 6a^2 + 8a \)
Correct Answer:
\( 10a^3 - 6a^2 + 8a \)
2 Part 2: Scaffolded FOIL Solutions (4 pts each • 8 pts total)
Criteria: 2 pts FOIL products, 1 pt combining like terms, 1 pt standard form
4. Multiply: \( (x + 6)(x + 2) \) 4 Points
F: \( x \cdot x = \mathbf{x^2} \)
O: \( x \cdot 2 = \mathbf{2x} \)
I: \( 6 \cdot x = \mathbf{6x} \)
L: \( 6 \cdot 2 = \mathbf{12} \)
Combined Terms: \( x^2 + (2x + 6x) + 12 = x^2 + 8x + 12 \)
Final Standard Form Answer:
\( x^2 + 8x + 12 \)
5. Multiply: \( (2x + 3)(x - 5) \) 4 Points
F: \( 2x \cdot x = \mathbf{2x^2} \)
O: \( 2x \cdot (-5) = \mathbf{-10x} \)
I: \( 3 \cdot x = \mathbf{3x} \)
L: \( 3 \cdot (-5) = \mathbf{-15} \)
Combined Terms: \( 2x^2 + (-10x + 3x) - 15 = 2x^2 - 7x - 15 \)
Final Standard Form Answer:
\( 2x^2 - 7x - 15 \)
💡 Pedagogical Tip for Page 1 Review:
Remind students that FOIL is simply a mnemonic for applying the distributive property twice. In Problem 5, highlight that the negative sign belongs to the 5: \( 2x(-5) = -10x \) and \( 3(-5) = -15 \). Common error: writing \( +15 \) or \( +10x \).
Polynomial Product Lab • Teacher Answer Key Page 1 of 2
POLYNOMIAL PRODUCT LAB — KEY | Page 2: Independent Practice & Modeling Solutions
Page 2 Subtotal: 16 Points
3 Part 3: Independent Practice Solutions (2 pts each • 8 pts total)
1 pt work/terms, 1 pt standard form answer
6. \( (3x - 2)(x + 4) \)
• FOIL: \( 3x^2 + 12x - 2x - 8 \)
