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# The Distributive Property

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1. The Distributive Property

2. The Distributive Property • The Distributive Property allows you to multiply each number inside a set of parenthesis by a factor outside the parenthesis and find the sum or difference of the resulting products. • To distribute means to separate or break apart and then dispense evenly. • Sometimes it is faster and easier to break apart a multiplication problem and use the distributive property to solve or simplify the problem using mental math strategies. • The distributive property is linked to factoring. When you factor problems, you identify what numbers or variables the problem has in common. When you distribute, you multiply the common numbers or variables to the numbers that have been grouped together.

3. The Distributive Property For any numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca; a(b - c) = ab - ac and (b - c)a = ba- ca; Distributive Property 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)

4. The Distributive Property Multiply 67  9 For any numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca; a(b - c) = ab - ac and (b - c)a = ba - ca; Or use the Distributive Property Add Multiply 67  9 Break apart the number 67 into (60 + 7) – the value of this number is still the same.

5. The Distributive Property Multiply 67  9 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; Or use the Distributive Property Multiply 67  9 Break apart the number 67 into (60 + 7) – the value of this number is still the same. Add 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)

6. The Distributive Property Multiply 48  7 For any numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca; a(b - c) = ab - ac and (b - c)a = ba - ca; Or use the Distributive Property Multiply 48  7 Break apart the number 48 into (50 - 2) – the value of this number is still the same. Subtract

7. The Distributive Property Multiply 6  473 For any numbers a, b, c, and d a(b + c + d) = ab + ac + ad Or use the Distributive Property Add Multiply 473  6 Break apart the number 473 into (400 + 70 + 3) – the value of this number is still the same.

8. The Distributive Property 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. Use the Distributive Property For any numbers a, b, and c a(b + c ) = ab + ac 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. Simplified

9. The Distributive Property • Term – a number (constant term), a variable (algebraic term), or a combination of numbers or variables that are added to form an expression. Given the problem 2x + 5, the terms are 2x and 5. Given the problem 2x – 5, the terms are 2x and –5. • Like terms are terms that share the same variable(s) and are raised to the same power. Remember that n’s go with n’s ; x’s go with x’s; n2 will only go with n2; numbers (constant term) by themselves go with numbers by themselves. Given the problem 2x + 5 + 3x + 2 + 4x2 + 5x2 can be simplified as 5x + 7 + 9x2. • Equivalent expression – Given the problem 5x + 4x; can be simplified to 9x. The expressions 5x + 4x and 9x are equivalent expressions because they name the same value. 9x is now in simplest form or the expression is said to be simplified.

10. The Distributive Property • Combining like terms – the process of adding or subtracting like terms. Given the problem 2x + 5 + 3x + 2 + 4x2 + 5x2 can be simplified as 5x + 7 + 9x2. The 2x and 3x can be combined to form 5x; the 5 and 2 can be combined to form 7, and the 4x2 and 5x2 can be combined to form 9x2. The simplified problem is then rewritten by placing the term with the highest exponent first, then the next term in decreasing order. 9x2 + 5x + 7 • Coefficient –a number and a letter is linked together by multiplication; the number or numerical factor is called the coefficient. Given the simplified algebraic expression 9x2 + 5x + 7; the 9 is the coefficient of the term 9x2, the 5 is the coefficient of the term 5x, and the 7 is referred as the constant term. • Note: All variables have a coefficient. Given the variable x; the coefficient is 1 because (1)(x) = x. Given the problem 2x + x + x; can be simplified as 2x + 1x + 1x = 4x.

11. The Distributive Property Simplify 4(7n + 2) +6 Notice the pattern: No symbol between the 4 and the parenthesis indicates a multiplication problem. Distribute by multiplication then perform the indicated operation inside the parenthesis. Use the Distributive Property For any numbers a, b, and c a(b + c ) = ab + ac Notice that 28n cannot be combined with any other n terms. The constant terms 8 and 6 are linked with addition and can be combined to form the constant number 14. The answer 28n + 14 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. Simplified

12. The Distributive Property Simplify 3(n + 2) + n Notice the pattern: No symbol between the 3 and the parenthesis indicates a multiplication problem. Distribute by multiplication then perform the indicated operation inside the parenthesis. Use the Distributive Property For any numbers a, b, and c a(b + c ) = ab + ac Notice that n has a coefficient of 1. After applying the distributive property – you can combine like terms. 3n and 1n can be combined to form 4n. The constant term 6 cannot be combined with any other constant terms. The answer 3n + 6 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. Simplified