The distributive properties of multiplication relative to addition and subtraction (also called distributive laws) are as follows: \[a(b+c)=ab+ac\] and \[(a+b)c=ac+bc\]
Examples
\[\begin{aligned}3(2a+5)&=3\cdot 2a+3\cdot 5\\ &=6a+15\\ \\ -3a^2(4a-7) &=(-3a^2)\cdot (4a)+(-3a^2)\cdot (-7)\\&=-3\cdot 4a^2\cdot a+-3\cdot -7\cdot a^2\\ &=-12a^3+21a^2\\ \\ (a-3)a&=a\cdot a+(-3)\cdot a\\ &=a^2-3a\end{aligned}\]
The properties \[a(b+c)=ab+ac\qquad\mathrm{and}\qquad (a+b)c=ac+bc\] are correct when you replace \(a\) and \(b\) by numbers and insert the omitted multiplication symbols in the right places, but they remain valid when letters represent algebraic expressions.
Of course you can also formulate distributive properties such as \[a(b-c)=ab-ac\qquad\mathrm{and}\qquad (a-b)c=ac-bc\] but these properties follow immediately from the former properties by reading \(-c\) and \(-b\) as \({}+(-c)\) and \({}+(-b)\), respectively.
You can also construct a multiplication table and, after filling out the table , add the two products.
| \[a(b+c)=ab+ac\] |
\[(a+b)c=ac+bc\] |
| \(\begin{array}{c|c|c}\cdot & b & c\\ \hline a &ab & ac \end{array}\) |
\(\begin{array}{c|c} \cdot & c\\ \hline a & ac \\ \hline b & bc \end{array}\) |
The distributive properties of multiplication relative to addition and subtraction are defined for the product of a single term with an expression consisting of two terms. But you can also do this for a second expression with three, four, or more terms.
| \[a(b+c+d)=ab+ac+ad\] |
\[a(b+c+d+e)=ab+ac+ad+ae\] |
| \(\begin{array}{c|c|c|c}\cdot & b & c & d\\ \hline a &ab & ac & ad \end{array}\) |
\(\begin{array}{c|c|c|c}\cdot & b & c & d & e\\ \hline a &ab & ac & ad & ae \end{array}\) |
The distributive properties help you expand brackets in the correct way. Examples illustrate how this goes.
Expand all brackets in \(6(6b+8)\).
Apply the distributive properties to: \[\begin{aligned}6(6b+8) &=(6\times 6b)+(6\times8)\\&=36b+48\end{aligned}\]
As the examples illustrate, you should be especially cautious when you are dealing with minus signs. An example where things can go wrong: \[-(a^2-a)\] Eliminating the brackets is a little tricky because the one-term is difficult to recognize; you should namely read the expression as \[-1\cdot(a^2-a)\] and then expansion of the brackets goes as follows: \[\begin{aligned}-(a^2-a) &=-1\cdot(a^2-a)\\ &=-1\cdot a^2\;+\;-1\cdot -a\\ &=-a^2+a\end{aligned}\]
Expanding/Removing Brackets (39:25)
Expansion of of single brackets
