Exploratory Exercise Kim was working on a problem in math when she ran across this problem. Distribute and simplify if possible. 2(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. 2(3x + 5) 1. Talk to your partner about what Kim s dad was trying to show. Then complete Kim s problem. In Lesson 7, you mainly used the commutative and associative properties, and only looked at examples of the distributive property. In this lesson we ll look closer at the distributive property and then in the next lesson you ll extend your knowledge about this important property. 2. 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 2? S.67
4. In each example below, one or more mistakes were made when distributing. Circle the mistakes and then write the correct expression. A. 2(3x + 5) = 6x + 5 B. 2 x(2x 3) = x 3 2 3 C. 3(3x + 4) = 9x + 4 D. 4(4x 5) = 16x 20 E. 2 2(4x + 6) = + 2 8x 12 F. 3 x(4x 5x+ 6) = 4 2 4 5 + 6 x 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? S.68
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. 2 (2x 3) + (7x+ 2) C. (3x+ 4) + ( 4x 7) D. (4x 5) + 2(3x+ 1) E. 2 2 2(4x + 6) + (7x 9x+ 3) F. 3 2 x(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. 2 2 (8x 9) (6x 2) B. (5x 2) (3x+ 9) C. 5( x+ 1) 6( x 1) D. 6x 5 (5x 6) E. 2 2 30x 20 2(10x 5x+ 7) F. 3 2 7 x x(8x + 9x 4) S.69
9. Determine which word matches each definition. Word Bank Polynomial Monomial Binomial Trinomial Degree of a Term Degree of a Polynomial Standard Form of a Polynomial Definition Word A. The sum of the exponents of the variable symbols that appear in the monomial B. A polynomial with only two terms C. The sum (or difference) of monomials D. The degree of the monomial term with the highest degree E. The terms are rewritten so that the terms are in order from greatest degree to least degree F. An algebraic expression generated using only the multiplication operator ( ). Thus, it does not contain + or operators. G. A polynomial with only three terms 10. Go back to Exercises 7 and 8 and write each polynomial in standard form, if it isn t already in standard form, and then determine the degree of the resulting polynomial. 11. Janie writes a polynomial expression using only one variable, xx, with degree 3. Max writes a polynomial expression using only one variable, xx, with degree 7. A. What can you determine about the degree of the sum of Janie s and Max s polynomials? B. What can you determine about the degree of the difference of Janie s and Max s polynomials? S.70
Homework Problem Set 1. Find each sum or difference by combining the parts that are alike. A. (4xx 2 + xx + 7) + (2xx 2 + 3xx + 1) B. (3xx 3 xx 2 + 8) (xx 3 + 5xx 2 + 4xx 7) C. 3(xx 3 + 8xx) 2(xx 3 + 12) D. (5 tt tt 2 ) + (9tt + tt 2 ) E. (3pp + 1) + 6(pp 8) (pp + 2) 2. Celina says that each of the following expressions is actually a binomial in disguise. For example, she sees that the expression in (i) is algebraically equivalent to 11aaaaaa 2aa 2, which is indeed a binomial. Is she right about the remaining four expressions? Explain your thinking. i. 5aaaaaa 2aa 2 + 6aaaaaa ii. 5xx 3 2xx 2 10xx 4 + 3xx 5 + 3xx ( 2)xx 4 iii. (tt + 2) 2 4tt iv. 5(aa 1) 10(aa 1) + 100(aa 1) v. (2ππππ ππππ 2 )ππ (2ππππ ππππ 2 ) 2ππ S.71
3. Suppose Janie writes a polynomial expression using only one variable, xx, with degree of 5, and Max writes a polynomial expression using only one variable, xx, with degree of 5. A. What can you determine about the degree of the sum of Janie s and Max s polynomials? B. What can you determine about the degree of the difference of Janie s and Max s polynomials? 4. Find each sum or difference. A. (2pp + 4) + 5(pp 1) (pp + 7) F. (12xx + 1) + 2(xx 4) (xx 15) B. (7xx 4 + 9xx) 2(xx 4 + 13) G. (13xx 2 + 5xx) 2(xx 2 + 1) C. (6 tt tt 4 ) + (9tt + tt 4 ) H. (9 tt tt 2 ) 3 2 (8tt + 2tt2 ) D. (5 tt 2 ) + 6(tt 2 8) (tt 2 + 12) I. (4mm + 6) 12(mm 3) + (mm + 2) E. (8xx 3 + 5xx) 3(xx 3 + 2) J. (15xx 4 + 10xx) 12(xx 4 + 4xx) S.72