2.4: The Distributive property of Multiplication over Addition and/or Subtraction

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The Distributive property of Multiplication over Addition and/or Subtraction

(0,8)(5,-5) (10,0)\(\overbrace{5\ \cdot\ (4}\ +\ 3)\ =(5\ \cdot \ 4\ )\ +\ (5\ \cdot \ 3 )\) (11.05,1.5)(0,1)1.5 (11.05,3.0)(1,0)5.5 (16.5,3.0)(0,-1)2.0 (10.1,-1.5)(1,0)3.2 (10.1,-0.5)(0,-1)1.0 (13.3,-0.5)(0,-1)1.0 (11.7,-1.5)(0,-1)1.0 (11.7,-2.5)(1,0)8.8 (20.4,-2.5)(0,1)2.2

\(5(4+3)=5(7)=35\)
and
\((5)(4)+(5)(3)=20+15=35\)
also.

The picture illustrates the Distributive property of multiplication over addition.

In general

\[\fbox{\Large \boldmath a(b+c)=ab+ac}\]
where \(a\), \(b\), and \(c\) are any real numbers.

A common mistake:

(0,4)(0,-5) (0,-2)\(x(y\pm z)=xy\pm z\) (0,-4)\(5(7+2)=5(7)+2\) (0,-6)\(5(9)=35+2\) (0,-8)\(45=37\) (0,-8)(1,1)7 (0,-1)(1,-1)7

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The picture illustrates the Distributive property of multiplication over Subtraction.

(0,8)(10,-4) (10,0)\(\overbrace{5\ \cdot\ (4\ }-\ 3)\ =(5\ \cdot \ 4\ )\ -\ (5\ \cdot \ 3 )\) (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.7 (15.0,-1.4)(0,1)1 (25.6,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)12.3

(11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0

\(5(4-3)=5(1)=5\) and \((5)(4)-(5)(3)=20-15=5\) also.

In general

\[\fbox{\large \boldmath a(b+c)=ab+ac}\]
[-15pt]

where \(a\), \(b\), and \(c\) are any real numbers.

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Waning:

You cannot distribute multiplication over multiplication.

(0,8)(10,-4) (10,0)\(\overbrace{5\ \cdot\ (4\ }\cdot \ 3)\ =(5\ \cdot \ 4\ )\ \cdot\ (5\ \cdot \ 3 )\) (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.3 (14.7,-1.4)(0,1)1 (24.4,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)11.1 (11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0 (10,-3)(5,2)15 (10,3)(5,-2)15

\(5(4\cdot3)=5(12)=60\) (Left side only)
and
\((5)(4)\cdot(5)(3)=20\cdot 15=300\) (Right side only)
The example shows that the distributive property does not apply to multiplication over multiplication.

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(0,8)(-10,-4) (9.8,0)\(\overbrace{16\ \div}\ (4\ \div\ 2)\ =(16\ \div \ 4\ )\ \div\ (16\ \div \ 2 )\) (14.2,-0.5)(0,-1)1 (14.2,-1.5)(1,0)8.7 (23.0,-1.5)(0,1)1 (10.9,1.5)(0,1)2 (10.9,2.5)(1,0)2.0 (12.9,2.5)(0,-1)1 (10.9,3.5)(1,0)4.5 (15.3,3.5)(0,-1)2.0

\(16\div (4\div 2)=16\div 2=8\)
and
\((16\div 4)\div(16\div 2)=4\div 8=\displaystyle \frac{1}{2}\)
which is different.

The example shows that the distributive property does not apply to division over division.

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