FOIL Calculator
Expand binomial products using the FOIL method with step-by-step algebraic analysis. Our algebra calculator supports polynomial expansion, special products, and comprehensive binomial multiplication studies.
Last updated: December 15, 2024
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Expression:
(x + 2) × (x + 3)
Algebraic Analysis
Final Result:
x^2 + 3x + 2x + 6
Method:
FOIL method expansion
Expanded Form:
x^2 + 3x + 2x + 6
FOIL Terms:
Combined:
x^2 + 3x + 2x + 6
Analysis:
The binomial product (x + 2)(x + 3) expands to x^2 + 3x + 2x + 6. The FOIL method systematically multiplies each term in the first binomial by each term in the second binomial.
Step-by-Step Solution:
- First: (x) × (x) = x^2
- Outer: (x) × (3) = 3x
- Inner: (2) × (x) = 2x
- Last: (2) × (3) = 6
- Add all terms: x^2 + 3x + 2x + 6
- Final result: x^2 + 3x + 2x + 6
FOIL Method:
- • First: Multiply the first terms of each binomial
- • Outer: Multiply the outer terms
- • Inner: Multiply the inner terms
- • Last: Multiply the last terms of each binomial
Quick Example Result
For (x + 2)(x + 3) using FOIL method:
x² + 5x + 6
First: x², Outer: 3x, Inner: 2x, Last: 6 → Combined: x² + 5x + 6
How This Calculator Works
Our FOIL calculator applies fundamental algebraic principles to expand binomial products systematically. The calculator uses the FOIL method(First, Outer, Inner, Last) to multiply two binomials and provides comprehensive algebraic analysis.
FOIL Method Formula
(a + b)(c + d) = ac + ad + bc + bdFirst: a × cOuter: a × dInner: b × cLast: b × dCombine like termsThe FOIL method ensures you multiply every term in the first binomial by every term in the second binomial. This systematic approach prevents missing any products and makes binomial multiplication reliable and consistent.
Visual representation of First, Outer, Inner, Last multiplication pattern
Algebraic Foundation
The FOIL method is based on the distributive property of multiplication over addition. When multiplying (a + b)(c + d), we distribute each term in the first binomial to each term in the second binomial. FOIL provides a memorable acronym to ensure all four necessary multiplications are performed systematically.
- First terms: multiply the first term of each binomial
- Outer terms: multiply the outer terms (first × last)
- Inner terms: multiply the inner terms (second × first)
- Last terms: multiply the last term of each binomial
Sources & References
- Algebra and Trigonometry - Robert Blitzer (6th Edition)Comprehensive treatment of polynomial operations and FOIL method
- National Council of Teachers of Mathematics - Algebra Education StandardsProfessional guidelines for teaching algebraic operations
- Khan Academy - Algebra BasicsEducational resources and practice problems for polynomial multiplication
Need help with other algebra calculations? Check out our quadratic formula calculator and polynomial calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Expression: (2x + 3)²
- Rewrite as: (2x + 3)(2x + 3)
- Method: FOIL expansion
- Pattern: Perfect square trinomial
FOIL Steps:
- First: (2x)(2x) = 4x²
- Outer: (2x)(3) = 6x
- Inner: (3)(2x) = 6x
- Last: (3)(3) = 9
- Combine: 4x² + 6x + 6x + 9
Result: 4x² + 12x + 9
This follows the perfect square pattern (a + b)² = a² + 2ab + b², where a = 2x and b = 3. Notice how the middle terms (Outer + Inner) give us 2ab = 2(2x)(3) = 12x, confirming the perfect square trinomial pattern.
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