Exploratory Exercise Kim was working on an exercise in math when she ran across this problem. Distribute and simplify if possible. (3x + 5) Kim s dad said, I remember doing something like this in school. He then drew two arcs on her paper. Distribute and simplify if possible. (3x + 5) 1. Talk to your partner about what Kim s dad was trying to show. Then complete Kim s problem.. What does the word distribute mean? Give two examples of the word in everyday use. 3. In math, distribute means to multiply out the parts of an expression. How does this definition relate to your definition from Exercise? Unit 3: Solving Equations & Inequalities S.41
4. In each example below, one or more mistakes were made when distributing. Circle the mistakes and then write the correct expression. A. (3x + 5) = 6x + 5 B. 3(x 3) = 6x 3 C. 3(3x + 4) = 9x + 4 D. 4(4x 5) = 16x 0 E. (4x + 6) = x + 8 1 F. 3 1(4 x 5x+ 6) = 3 4 5 6 x x+ 5. What was the common mistake made in 4A, 4B and 4C? 6. What was the common mistake made in 4D, 4E and 4F? Unit 3: Solving Equations & Inequalities S.4
ADDING POLYNOMIALS 7. Add the following polynomials by combining like terms. Be careful you will have to distribute in a few of them. A. (3x+ 5) + (7x 3) B. (x 3) + (7x+ ) C. (3x+ 4) + ( 4x 7) D. (4x 5) + (3x+ 1) E. (4x + 6) + (7x 9x+ 3) F. 3 (4x 5x+ 6) + (6x + 7x 3) SUBTRACTING POLYNOMIALS When subtracting polynomials, you will need to distribute the negative sign to all the terms in the parentheses. 8. Subtract these polynomials and then combine like terms. A. (8x 9) (6x ) B. (5x ) (3x+ 9) C. 5( x+ 1) 6( x 1) D. 6x 5 (5x 6) E. 30x 0 (10x 5x+ 7) F. 3 7 x (8x + 9x 4) Unit 3: Solving Equations & Inequalities S.43
Distributing Fractions 9. Distribute and then combine like terms to simplify each expression. Remember, you aren t solving equations. These are all expressions. A. 1 3 ( x 4 x+ 7 ) B. ( 9 x + 3 x ) 3 1 C. ( ) 4 3 15 5 6 8 5 x + 4x+ 8 + x 4x D. ( x+ ) + ( x ) 3 4 4 16 3 x x x 14 x 14 + + + + F. ( x ) E. ( ) 4 1 18 3 3 6 1 + + x 6 6 Unit 3: Solving Equations & Inequalities S.44
Homework Problem Set Find each sum or difference. 1. ( 3x 4) ( 5x 7) +. ( 6x 1) ( x + 8) 3. ( 1x 9) ( 7x 3) ( 6x 1) + + 4. (4xx + xx + 7) + (xx + 3xx + 1) 5. (3xx 3 xx + 8) (xx 3 + 5xx + 4xx 7) 6. 3(xx 3 + 8xx) (xx 3 + 1) 7. (5 tt tt ) + (9tt + tt ) 8. (3pp + 1) + 6(pp 8) (pp + ) 9. (pp + 4) + 5(pp 1) (pp + 7) 10. (6 tt tt 4 ) + (9tt + tt 4 ) Unit 3: Solving Equations & Inequalities S.45
11. (7xx 4 + 9xx) (xx 4 + 13) 1. (5 tt ) + 6(tt 8) (tt + 1) 13. (8xx 3 + 5xx) 3(xx 3 + ) 14. (1xx + 1) + (xx 4) (xx 15) 15. (13xx + 5xx) (xx + 1) 16. (9 tt tt ) 3 (8tt + tt ) 17. (4mm + 6) 1(mm 3) + (mm + ) 18. (15xx 4 + 10xx) 1(xx 4 + 4xx) 19. CHALLENGE Celina says that each of the following expressions is actually a -term expression (called a binomial) in disguise. For example, she sees that the expression in (i) is algebraically equivalent to 11aaaaaa aa, which is indeed a -term expression. Is she right about the remaining four expressions? Explain your thinking. i. 5aaaaaa aa + 6aaaaaa ii. 5xx 3 xx 10xx 4 + 3xx 5 + 3xx ( )xx 4 iii. (tt + ) 4tt iv. 5(aa 1) 10(aa 1) + 100(aa 1) v. (ππππ ππππ )ππ (ππππ ππππ ) ππ Unit 3: Solving Equations & Inequalities S.46