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Multiplying Binomials

Multiply (ax+b)(cx+d) using FOIL: First, Outer, Inner, Last. (a+b)² = a²+2ab+b². (a−b)² = a²−2ab+b². (a+b)(a−b) = a²−b² — conjugate pairs cancel the middle term.

Concept Fundamentals
ac+ad+bc+bd
FOIL
a²+2ab+b²
(a+b)²
a²−2ab+b²
(a−b)²
a²−b²
(a+b)(a−b)

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FOIL is the distributive property applied twice: (a+b)(c+d) = ac+ad+bc+bd. (x+4)(x−4) = x²−16 — conjugate pair makes middle terms cancel. Area model: rectangle (ax+b)×(cx+d) splits into four sub-rectangles — one per FOIL term.

Key quantities
ac+ad+bc+bd
FOIL
Key relation
a²+2ab+b²
(a+b)²
Key relation
a²−2ab+b²
(a−b)²
Key relation
a²−b²
(a+b)(a−b)
Key relation

Ready to run the numbers?

Why: Binomial multiplication is foundational for factoring, solving quadratics, and mental math. Conjugate pairs (a+b)(a−b)=a²−b² simplify multiplication: 21×19=(20+1)(20−1)=399.

How: FOIL: First (a·c), Outer (a·d), Inner (b·c), Last (b·d). Combine like terms. Special: (a+b)² middle = 2ab; (a−b)² middle = −2ab; (a+b)(a−b) middle cancels. Triple: multiply first two, then by third.

FOIL is the distributive property applied twice: (a+b)(c+d) = ac+ad+bc+bd.(x+4)(x−4) = x²−16 — conjugate pair makes middle terms cancel.
Sources:Khan Academy — Multiplying BinomialsWolfram MathWorld — FOIL

Run the calculator when you are ready.

Expand BinomialsFOIL, special products, triple

📐 Examples — Click to Load

First Binomial (ax + b) × Second (cx + d)

binomial.sh
CALCULATED
$ multiply --a=1 --b=3 --c=1 --d=4
Product
(x + 3)(x + 4) = x^2 + 7x + 12
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Term Magnitudes

Contribution by Term

📐 Calculation Steps

Multiplying: (x + 3)(x + 4)
Using FOIL: First, Outer, Inner, Last
1. First: x × x = x^2
2. Outer: x × 4 = 4x
3. Inner: 3 × x = 3x
4. Last: 3 × 4 = 12
5. Combine: x^2 + 4x + 3x + 12
Result: (x + 3)(x + 4) = x^2 + 7x + 12

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

⚡

21×19 = (20+1)(20−1) = 400−1 = 399. Conjugate pairs simplify mental multiplication.

— Mental Math

📊

The area model: a rectangle (ax+b)×(cx+d) splits into four smaller rectangles — one for each FOIL term.

— Geometry

📋 Key Takeaways

  • • FOIL = First, Outer, Inner, Last — multiply (ax+b)(cx+d) term by term
  • • (a+b)² = a² + 2ab + b² — perfect square of a sum
  • • (a−b)² = a² − 2ab + b² — perfect square of a difference
  • • (a+b)(a−b) = a² − b² — difference of squares, middle terms cancel
  • • Area model: FOIL corresponds to four rectangles in a 2×2 grid

💡 Did You Know?

📐FOIL is the distributive property applied twice: (a+b)(c+d) = a(c+d) + b(c+d) = ac+ad+bc+bd.Source: Algebra
🔢(x+4)(x-4) = x²-16 — the conjugate pair makes the middle terms cancel. Use this for mental math!Source: Special Products
⚡21×19 = (20+1)(20-1) = 400−1 = 399. Conjugate pairs simplify multiplication.Source: Mental Math
📊The area model: a rectangle (ax+b)×(cx+d) splits into four smaller rectangles — one for each FOIL term.Source: Geometry
🎯(x+5)² = x² + 10x + 25. The middle coefficient is always 2×(first)×(last) = 2·x·5.Source: Perfect Square
🧮Triple product (x+1)(x+2)(x+3) expands to x³+6x²+11x+6. Multiply in pairs.Source: Extension

📖 How It Works

To multiply (ax+b)(cx+d), apply FOIL: First (a·c·x²), Outer (a·d·x), Inner (b·c·x), Last (b·d). Combine like terms (the x terms) to get the final polynomial.

Example: (x+3)(x+4)

F: x·x = x². O: x·4 = 4x. I: 3·x = 3x. L: 3·4 = 12. Sum: x² + 4x + 3x + 12 = x² + 7x + 12.

Special: (x+4)(x-4)

F: x². O: -4x. I: +4x. L: -16. Outer + Inner cancel! Result: x² − 16.

Geometric: Area model

Draw a rectangle with sides (ax+b) and (cx+d). The four sub-rectangles are ac·x², ad·x, bc·x, and bd. Total area = FOIL result.

🎯 Expert Tips

💡 Spot Conjugates

(a+b)(a−b) = a²−b². Same terms, opposite signs — middle cancels.

💡 Perfect Squares

(a+b)² middle term = 2ab. (a−b)² middle term = −2ab.

💡 Combine Like Terms

After FOIL, always add Outer + Inner (the x terms).

💡 Triple Product

Multiply first two binomials, then multiply result by the third.

📊 Special Products Table

PatternFormulaWhen to Use
FOIL(a+b)(c+d) = ac+ad+bc+bdAny two binomials
Sum squared(a+b)² = a²+2ab+b²Same binomial, both +
Diff squared(a−b)² = a²−2ab+b²Same binomial, both −
Diff of squares(a+b)(a−b) = a²−b²Same terms, opposite signs

❓ FAQ

What is FOIL?

FOIL = First, Outer, Inner, Last. A mnemonic for multiplying (ax+b)(cx+d): multiply First terms, Outer terms, Inner terms, Last terms, then combine.

When do middle terms cancel?

In (a+b)(a−b): Outer = ab, Inner = −ab. They cancel, leaving a²−b² (difference of squares).

How do I multiply (x+5)²?

Use (a+b)² = a²+2ab+b²: x² + 2·x·5 + 25 = x² + 10x + 25.

What about (x+1)(x+2)(x+3)?

First multiply (x+1)(x+2) = x²+3x+2. Then multiply by (x+3) to get x³+6x²+11x+6.

What is the area model?

A rectangle with sides (ax+b) and (cx+d) splits into four rectangles. Their areas are the four FOIL products. Total area = expanded form.

How does this relate to factoring?

Multiplying binomials (expand) is the inverse of factoring. Use Factoring Trinomials to go from x²+5x+6 back to (x+2)(x+3).

📝 Quick Reference

F
First: a×c
O
Outer: a×d
I
Inner: b×c
L
Last: b×d

📐 Area Model Visualization

FOIL can be visualized with an area model. Draw a rectangle with sides (ax+b) and (cx+d). Split it into four smaller rectangles:

ac·x² (First)ad·x (Outer)
bc·x (Inner)bd (Last)

Total area = ac·x² + ad·x + bc·x + bd = combined FOIL result.

This geometric view reinforces why we multiply each term in the first binomial by each term in the second — we are finding the area of each sub-rectangle.

🔄 FOIL vs Distributive Property

FOIL is a special case of the distributive property. For (a+b)(c+d), we distribute twice:

(a+b)(c+d) = (a+b)·c + (a+b)·d = ac + bc + ad + bd

Reordering: ac + ad + bc + bd = First + Outer + Inner + Last

The same idea extends to (ax+b)(cx²+dx+e) — multiply each term in the first by each in the second, then combine like terms.

🧠 Mental Math: Conjugate Pairs

Use (a+b)(a−b) = a²−b² for quick mental multiplication. Example: 47×53 = (50−3)(50+3) = 50²−3² = 2500−9 = 2491. Or 21×19 = (20+1)(20−1) = 400−1 = 399.

Strategy: Find the midpoint m, write as (m+d)(m−d) = m²−d²

This works whenever the two numbers are equidistant from a round number. Try 34×26 = (30+4)(30−4) = 900−16 = 884.

⚠️ Disclaimer: This calculator is for educational use. FOIL applies to binomials. For longer polynomials, use the distributive property (each term × each term).

Multiplying binomials is the inverse of factoring. Use the Factoring Trinomials Calculator to go from expanded form back to binomials.

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