Distributive Property: Expand and Factor Algebraic Expressions with Examples

Grade 6 examples and questions on how to use the distributive property in algebra with detailed solutions and explanations are presented.

Use the Distributive Property to Expand Algebraic Expressions

For real numbers \(a\), \(b\), and \(c\), the distributive property is given by:

\[ \Large {\color{red}{a(b+c) = ab + ac}} \]

Let’s evaluate the expression \(2(3+4)\) in two different ways.

\[ 2(3+4) = 2(7) = 14 \]

\[ 2(3+4) = (2)(3) + (2)(4) = 6 + 8 = 14 \]

Both methods give the same result. However, when variables are involved, the distributive property is essential.

For example: \[ 2(x+6) = (2)(x) + (2)(6) = 2x + 12 \]

Since \(x\) is a variable, you cannot simplify \(x+6\) directly; the distributive property must be used.

The distributive property can also be applied in reverse to factor expressions.

\[ \Large {\color{red}{ab + ac = a(b+c)}} \]

\[ 3x+6 = 3x + 3(2) = 3(x+2) \]

\(3(x+1)+3\)
\(5(1+n)+6\)
\(5(a+2)+2(a+3)\)
\(2(1+b)+6(b+2)+4\)
\(2(a+b)+3(a+b)\)

\(2x+4\)
\(3x+3\)
\(4a+12\)
\(21+7b\)
\(15+5x\)
\(\dfrac{x}{2}+\dfrac{1}{2}\)

\(2(x+2) = 2x+4\)
\(3(a+4) = 3a+12\)
\(4(3+b) = 12+4b\)
\(5(3+n) = 15+5n\)
\(2(a+b) = 2a+2b\)
\(2(x+y+4) = 2x+2y+8\)
\((7+b)4 = 28+4b\)

\(3(x+1)+3 = 3x+3+3 = 3x+6\)
\(5(1+n)+6 = 5+5n+6 = 5n+11\)
\(5(a+2)+2(a+3) = 5a+10+2a+6 = 7a+16\)
\(2(1+b)+6(b+2)+4 = 2+2b+6b+12+4 = 8b+18\)
\(2(a+b)+3(a+b) = 2a+2b+3a+3b = 5a+5b\)

\(2x+4 = 2(x+2)\)
\(3x+3 = 3(x+1)\)
\(4a+12 = 4(a+3)\)
\(21+7b = 7(3+b)\)
\(15+5x = 5(3+x)\)
\(\dfrac{x}{2}+\dfrac{1}{2} = \dfrac{1}{2}(x+1)\)