Practice Problem #1

• 7p + q – 7(q + 6p)

• - (a – b) + 2(a + b)

• 3(-2n – 5) – 2(-3n + 5)

Practice Problem #2

• Eighty students from a school went to the taping of a television program. They filled 7 rows of seats and there were 3 students in the eighth row. How many seats were in each row??

Recall the distributive property of multiplication over addition . . .

symbolically:

a × (b + c) = a × b + a × cand pictorially (rectangular array area model):

a × b a × ca

b c

An example: 6 x 13 using your mental math skills . . .

symbolically:

6 × (10 + 3) = 6 × 10 + 6 × 3and pictorially (rectangular array area model):

6 × 10 6 × 36

10 3

What about 12 x 23? Mental math skills?

(10+2)(20+3) = 10×20 + 10×3 + 2×20 + 2×3

10 × 20 10 × 310

20 3

2 × 32 2 × 20

200

30

40

+ 6

276

Distributive Property

Distributive Property

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

When a number or letter is separated by parentheses and there are no other operation symbols – it means to distribute by multiplying the numbers or variables together.

Find the sum (add) or difference (subtract) of the distributed products.

Notice that it doesn’t matter which side of the expression the letter a is written on because of the symmetric property which states for any real numbers a and b; if a = b, then b = a.

If a(b + c) = ab + ac, then ab + ac = a(b + c)

Notice that it doesn’t matter which side of the expression the letter a is written on because of the symmetric property which states for any real numbers a and b; if a = b, then b = a.

If a(b + c) = ab + ac, then ab + ac = a(b + c)

And now for multiplying binomials

(a+b)×(c+d) = a×(c+d) + b×(c+d)

= a×c + a×d + b×c + b×d

a × c a × da

c d

b × db b × c

Or use the Distributive Property

Or use the Distributive Property

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

Multiply 67 9

3

9

67

6

60

Multiply 67 9

Break apart the number 67 into (60 + 7) – the value of this number is still the same.

Multiply 67 9

Break apart the number 67 into (60 + 7) – the value of this number is still the same.

7,60,9 now cba

)760(9 540 63

603AddAdd

Or use the Distributive Property

Or use the Distributive Property

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

Multiply 67 9

3

9

67

6

60Multiply 67 9

Break apart the number 67 into (60 + 7) – the value of this number is still the same.

Multiply 67 9

Break apart the number 67 into (60 + 7) – the value of this number is still the same.

7,60,9 now cba

9)760( 540 63

603AddAdd

Notice that it doesn’t matter which side of the expression the letter a is written on because of the symmetric property which states for any real numbers a and b; if a = b, then b = a.

If a(b + c) = ab + ac, then ab + ac = a(b + c)

Notice that it doesn’t matter which side of the expression the letter a is written on because of the symmetric property which states for any real numbers a and b; if a = b, then b = a.

If a(b + c) = ab + ac, then ab + ac = a(b + c)

Or use the Distributive Property

Or use the Distributive Property

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

For any numbers a, b, and c,

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

Multiply 48 7

6

7

48

5

33

Multiply 48 7

Break apart the number 48 into (50 - 2) – the value of this number is still the same.

Multiply 48 7

Break apart the number 48 into (50 - 2) – the value of this number is still the same.

2,50,7 now cba

)250(7 350 14

336 SubtractSubtract

Or use the Distributive Property

Or use the Distributive Property

For any numbers a, b, c, and d

a(b + c + d) = ab + ac + ad

For any numbers a, b, c, and d

a(b + c + d) = ab + ac + ad

Multiply 6 473

8

6

473

1

Multiply 473 6

Break apart the number 473 into (400 + 70 + 3) – the value of this number is still the same.

Multiply 473 6

Break apart the number 473 into (400 + 70 + 3) – the value of this number is still the same.

3,70,400,6 now dcba

)370400(6 2400420

2838AddAdd

8 3

6

473

1 4

8 283

6

473

1 4

18

Use the Distributive Property

Use the Distributive Property

For any numbers a, b, and c

a(b + c ) = ab + ac

For any numbers a, b, and c

a(b + c ) = ab + ac

Simplify 5(3n + 4)Notice the pattern: No symbol between the 5 and the parenthesis indicates a multiplication problem. Distribute by multiplication then perform the indicated operation inside the parenthesis.

Notice that 15n means (15)(n) and is linked by multiplication and that the number 20 is by itself. These two terms are not alike and therefore cannot be combined. The answer 15n + 20 is simplified because we do not know what the value of n is at this time and cannot complete the multiplication part of this problem.

Notice that 15n means (15)(n) and is linked by multiplication and that the number 20 is by itself. These two terms are not alike and therefore cannot be combined. The answer 15n + 20 is simplified because we do not know what the value of n is at this time and cannot complete the multiplication part of this problem.

4,3,5 now cnba

)43(5 n )3(5 n

)4(5

n15 20SimplifiedSimplified

We note that the product of the two binomials has four terms – each of these is a partial product. We multiply each term of the first binomial by each term of the second binomial to get the four partial products.

Product of the FIRST terms of the binomials

Product of the OUTSIDE terms of the binomials

Product of the INSIDE terms of the binomials

Product of the LAST terms of the binomials

F + O + I + L

( a + b )( c + d ) = ac + ad + bc + bd

Because this product is composed of the First, Outside, Inside, and Last terms, this pattern is often referred to as FOIL method of multiplying two binomials. Note that each of these four partial products represents the area of one of the four rectangles making up the large rectangle.