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๐Ÿ“ฆ

Distributive Property

a(b+c)=ab+ac: multiply each term inside by the factor outside. The foundation of algebraic expansion. Reverse: ab+ac=a(b+c) factors out the GCF. FOIL extends this to (a+b)(c+d).

Concept Fundamentals
ab+ac
a(b+c)
ac+ad+bc+bd
(a+b)(c+d)
a(b+c)
ab+ac
Greatest common factor
GCF

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Distributive property is one of the fundamental axioms of algebra. FOIL is the distributive property applied twice to binomials. Factoring reverses expansion; GCF is the common factor of all terms.

Key quantities
ab+ac
a(b+c)
Key relation
ac+ad+bc+bd
(a+b)(c+d)
Key relation
a(b+c)
ab+ac
Key relation
Greatest common factor
GCF
Key relation

Ready to run the numbers?

Why: The distributive property underlies all polynomial multiplication and factoring. Mental math: 7ร—23=7(20+3)=140+21=161. Area model: a(b+c) is the sum of two rectangles.

How: Expand: multiply the outside term by each term inside, then add. Factor: find the GCF of all terms, write it outside parentheses, divide each term by it. FOIL: First, Outer, Inner, Last.

Distributive property is one of the fundamental axioms of algebra.FOIL is the distributive property applied twice to binomials.
Sources:Khan Academy โ€” Distributive PropertyWolfram MathWorld โ€” Distributive

Run the calculator when you are ready.

Expand and FactorDistributive property, FOIL, GCF factoring

๐Ÿ“ฆ Examples โ€” Click to Load

(ax + b)(cx + d)

distribute.sh
CALCULATED
$ distributive --expand
Original
(x+2)(x+3)
Expanded
xยฒ + 5x + 6
Factored
(x+2)(x+3)
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๐Ÿ“ Calculation Steps

  1. Original: (x+2)(x+3)
  2. FOIL: First 1ร—1=1xยฒ, Outer 1ร—3=3x, Inner 2ร—1=2x, Last 2ร—3=6
  3. Combine: 3x + 2x = 5x
  4. Expanded: xยฒ + 5x + 6

Term Magnitudes

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

๐Ÿงฎ Fascinating Math Facts

๐Ÿ“

Area model: a(b+c) = area of rectangle aร—(b+c) = two rectangles ab + ac.

โ€” Geometry

๐Ÿ”ข

3(4+5)=3ร—4+3ร—5=12+15=27. Same as 3ร—9=27.

โ€” Verify

๐Ÿ“‹ Key Takeaways

  • โ€ข The distributive property states: a(b + c) = ab + ac โ€” multiply each term inside by the factor outside
  • โ€ข It is a fundamental property of real numbers โ€” one of the core axioms of algebra
  • โ€ข FOIL (First, Outer, Inner, Last) is the distributive property applied to (a+b)(c+d)
  • โ€ข Factoring reverses distribution: ab + ac = a(b + c) by finding the GCF
  • โ€ข The area model visualizes distribution: a rectangle with sides (a) and (b+c) has area ab + ac

๐Ÿ’ก Did You Know?

๐Ÿ“The area model: a(b+c) is the area of a rectangle with width a and length (b+c) โ€” split into two rectangles ab and acSource: Visual Algebra
๐Ÿ”—FOIL is just distribution: (a+b)(c+d) = ac + ad + bc + bd โ€” four partial productsSource: Algebra I
๐ŸงฎMental math: 7ร—98 = 7(100โˆ’2) = 700 โˆ’ 14 = 686. Distribution makes hard multiplication easy!Source: Number Sense
โž–a(bโˆ’c) = ab โˆ’ ac. Distribution works with subtraction โ€” the negative distributes tooSource: Signed Numbers
๐Ÿ“ŠFactoring is the inverse: 6x+9 = 3(2x+3). GCF goes outside the parenthesesSource: Polynomials
๐ŸŽฏNested: a(b(c+d)) = a(bc+bd) = abc+abd. Distribute from the inside outSource: Advanced Algebra

๐Ÿ“– How the Distributive Property Works

To expand a(b + c), multiply the factor a by each term inside the parentheses: aร—b + aร—c = ab + ac. Think of it as "distributing" the multiplication over the addition. For (a+b)(c+d), apply distribution twice โ€” or use FOIL: First (ac), Outer (ad), Inner (bc), Last (bd).

Example: 3(x + 4)

3(x + 4) = 3ยทx + 3ยท4 = 3x + 12. Each term gets multiplied by 3.

Example: (x+2)(x+3)

FOIL: xยทx + xยท3 + 2ยทx + 2ยท3 = xยฒ + 3x + 2x + 6 = xยฒ + 5x + 6.

๐ŸŽฏ Expert Tips

๐Ÿ’ก Watch the Signs

โˆ’2(3xโˆ’5) = โˆ’6x + 10. Negative times negative = positive.

๐Ÿ’ก Mental Math Shortcut

Break numbers: 23ร—12 = 23(10+2) = 230 + 46 = 276.

๐Ÿ’ก Verify by Substituting

Check: 3(x+4) and 3x+12 give same value for any x.

๐Ÿ’ก Factor by GCF

6x+9 = 3(2x+3). Find the greatest common factor first.

๐Ÿ“Š Reference: Distribution Rules

FormExpansion
a(b + c)ab + ac
a(b โˆ’ c)ab โˆ’ ac
(a+b)(c+d)ac + ad + bc + bd
โˆ’a(b + c)โˆ’ab โˆ’ ac

โ“ FAQ

What is the distributive property?

a(b+c) = ab + ac. A number times a sum equals the sum of that number times each term. It is a fundamental property of real numbers.

How do I expand (a+b)(c+d)?

Use FOIL: First (ac), Outer (ad), Inner (bc), Last (bd). Result: ac + ad + bc + bd. Same as distributing twice.

What about negative factors?

โˆ’a(b+c) = โˆ’ab โˆ’ ac. The negative distributes to each term. โˆ’2(3xโˆ’5) = โˆ’6x + 10.

How do I factor an expression?

Find the GCF of all terms. Write it outside parentheses with the remaining terms inside. 6x+9 = 3(2x+3).

What is the area model?

A rectangle with sides a and (b+c) has area a(b+c). Splitting into two rectangles gives areas ab and ac โ€” visualizing distribution.

Can I use distribution for mental math?

Yes! 7ร—98 = 7(100โˆ’2) = 700โˆ’14 = 686. Break numbers into easier parts.

๐Ÿ“ Quick Reference

a(b+c)
= ab + ac
FOIL
First Outer Inner Last
GCF
Factor out common
โˆ’a(b+c)
= โˆ’ab โˆ’ ac

โš ๏ธ Disclaimer: This calculator handles numeric and simple algebraic distribution. For complex polynomials with variables, manual verification is recommended. Educational use only.

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