Distributive Law and Algebraic Identities

What you’ll learn

Expand brackets and recognise standard products.

ពន្លាតវង់ក្រចក និងស្គាល់ផលគុណស្តង់ដារ។

Meaning and use

The distributive law multiplies every term inside a bracket. Squaring a sum includes two cross-products, so it is not usually the sum of the squares. An identity holds for all values in its domain, unlike an equation solved for particular values.

ច្បាប់បំបែកគុណត្រូវគុណគ្រប់តួក្នុងវង់ក្រចក។ ការ៉េនៃផលបូកមានតួកណ្ដាល 2ab ដូច្នេះមិនស្មើផលបូកការ៉េទាំងពីរជាទូទៅទេ។

Formulas

Distributive law

\[a(b+c)=ab+ac\]

Common identities

\[(a+b)^2=a^2+2ab+b^2,\qquad (a+b)(a-b)=a^2-b^2\]

Symbols and units

Symbol	Meaning	Unit or restriction
a, b, c	Numbers or algebraic expressions	Compatible units if quantities are added
2ab	The two combined cross-products	Same algebraic degree as a² and b²

Conditions and limits

• Apply multiplication to every term, including signs.
• Do not distribute a square over addition.
• ច្បាប់ត្រូវរក្សាសញ្ញាអវិជ្ជមានពេលពន្លាត។

Worked examples

Expanding a bracket

Expand \(3(x+4)\).

1. Multiply 3 by x and by 4.
2. Obtain \(3x+12\).

\(3x+12\)

ត្រូវគុណទាំង x និង 4។

Squaring a sum

Expand \((x+5)^2\).

1. Use \((a+b)^2=a^2+2ab+b^2\).
2. Substitute a = x and b = 5.
3. Obtain \(x^2+10x+25\).

\(x^2+10x+25\)

តួកណ្ដាលកើតពីផលគុណ x×5 ពីរដង។

Common mistakes

Instead of	Use	Why
(a + b)² = a² + b²	(a + b)² = a² + 2ab + b²	Include the cross-products.
−2(x − 3) = −2x − 6	−2(x − 3) = −2x + 6	A negative times a negative is positive.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Expand 4(x + 2).
2. Expand (x + 3)².
3. Simplify (x + 7)(x − 7).

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 4x + 8

   Distribute 4 to both terms.

2. x² + 6x + 9

   Use the squared-sum identity.

3. x² − 49

   Use the difference-of-squares identity.

Quick recap

• Distribute to every term.
• Keep the middle term in a squared sum.
• Check signs with a numerical substitution.