7.3 Arithmetic Operations with Algebraic Expressions

All arithmetic operations can be applied to algebraic expressions by following the rules that we have learned thus far, including the order of operations (BEDMAS), properties of exponents, and operations with signed numbers.

Addition and Subtraction

Addition and Subtraction of Monomials

Addition and subtraction of monomials can be performed by adding and subtracting the coefficients of like terms, according to the rules of signed numbers.

Note: Recall that if a coefficient of a term is not written, it is 1.

Example 7.3-a: Adding and Subtracting Monomials

1. Add [latex] 6x [/latex] and [latex] 3x [/latex]

   
   

2. Add [latex] 4x^2y [/latex] and [latex] x^2y [/latex]

   
   

3. Subtract [latex] 5x^3 [/latex] from [latex] 7x^3 [/latex]

   
   

4. Subtract [latex] 8x [/latex] from the sum of [latex] 7x [/latex] and [latex] 4x [/latex]

   
   

5. Add [latex] 5x [/latex] and [latex] 6y [/latex]

   
   

6. Subtract [latex] 2y^2 [/latex] from [latex] 7y^3 [/latex]

   
   

Solution

1. [latex] 6x + 3x [/latex]   Adding like terms,

   [latex] = 9x [/latex]

   
   

2. [latex] 4x^2y + x^2y [/latex]   Adding like terms,

   [latex] = 5x^2y [/latex]

   
   

3. [latex] 7x^3 - 5x^3 [/latex]   Subtracting like terms,

   [latex] = 2x^3 [/latex]

   
   

4. [latex] (7x + 4x) - (8x) [/latex]   Adding like terms inside the brackets,

   [latex] = 11x - 8x [/latex]   Subtracting like terms,

   [latex] = 3x [/latex]

   
   

5. [latex] 5x + 6y [/latex]   Since these are not like terms, we cannot simplify the expression at all.

   
   

6. [latex] 7y^3 - 2y^2 [/latex]   Since these are not like terms, we cannot simplify the expression at all.

   
   

Addition and Subtraction of Polynomials

When adding or subtracting algebraic expressions, first collect the like terms and group them, then add or subtract the coefficients of the like terms.

• For addition of polynomials (indicated by a plus '+' sign outside the brackets), brackets can be removed without changing any of the signs of the terms within the brackets.
• For subtraction of polynomials (indicated by a negative ‘–’ sign outside the brackets), brackets can be removed by distributing the negative sign to the terms within the brackets; this is equivalent to multiplying every term within the brackets by –1, resulting in the signs changing on each term within the brackets.

Example 7.3-b: Adding and Subtracting Algebraic Expressions

Evaluate the following expressions:

1. Add [latex] (3x + 7) [/latex] and [latex] (5x + 3) [/latex]

   
   

2. Add [latex] (4y^2 - 8y - 9) [/latex] and [latex] (2y^2 + 6y - 2) [/latex]

   
   

3. Subtract [latex] (x^2 + 5x - 7) [/latex] from [latex] (2x^2 - 2x + 3) [/latex]

   
   

4. Subtract [latex] [5x - (x + 8)] [/latex] from [latex] (x - 3) [/latex]

   
   

Solution

1. [latex] (3x + 7) + (5x + 3) [/latex]   Removing the brackets,

   [latex] = 3x + 7 + 5x + 3 [/latex]   Grouping like terms,

   [latex] = 3x + 5x + 7 + 3 [/latex]   Adding like terms,

   [latex] = 8x + 10 [/latex]

   
   

2. [latex] (4y^2 - 8y - 9) + (2y^2 + 6y - 2) [/latex]   Removing the brackets,

   [latex] = 4y^2 - 8y - 9 + 2y^2 + 6y - 2 [/latex]   Grouping like terms,

   [latex] = 4y^2 + 2y^2 - 8y + 6y - 9 - 2 [/latex]   Adding and subtracting like terms,

   [latex] = 6y^2 - 2y - 11 [/latex]

   
   

3. [latex] (2x^2 - 2x + 3) - (x^2 + 5x - 7) [/latex]   Removing the brackets by distributing the negative sign to all the terms within the bracket,

   [latex] = 2x^2 - 2x + 3 - x^2 - 5x + 7 [/latex]   Grouping like terms,

   [latex] = 2x^2 - x^2 - 2x - 5x + 3 + 7 [/latex]   Adding and subtracting like terms,

   [latex] = x^2 - 7x + 10 [/latex]

   
   

4. [latex] (x - 3) - [5x - (x + 8)] [/latex]   Removing the brackets by distributing the negative sign to all the terms within the bracket,

   [latex] = x - 3 - [5x - x - 8] [/latex]

   [latex] = x - 3 - 5x + x + 8 [/latex]   Grouping like terms,

   [latex] = x - 5x + x - 3 + 8 [/latex]   Adding and subtracting like terms,

   [latex] = -3x + 5 [/latex]

   
   

Multiplication

Multiplying a Monomial by a Monomial

Multiplying a monomial by another monomial is just simplifying an algebraic term, as we did in the previous section. Multiply the coefficients together and multiply the variables together using the properties of exponents, where applicable.

Example 7.3-c: Multiplying Monomials by Monomials

Evaluate the following expressions:

1. Multiply [latex] 6x^2y [/latex] and [latex] 5xy [/latex]

   
   

2. Multiply [latex] (3a^3) [/latex], [latex] (-4ab) [/latex] and [latex] (2b^2) [/latex]

   
   

Solution

1. [latex] (6x^2y)(5xy) [/latex]

   [latex] = (6)(5)(x^2)(x)(y)(y) [/latex]

   [latex] = 30x^3y^2 [/latex]

   
   

2. [latex] (3a^3)(-4ab)(2b^2) [/latex]

   [latex] = (3)(-4)(2)(a^3)(a)(b)(b^2) [/latex]

   [latex] = -24a^4b^3 [/latex]

   
   

Multiplying a Polynomial by a Monomial

When multiplying a polynomial by a monomial, multiply the monomial by each term of the polynomial. This is also known as the distributive property of multiplication, as shown below.

[latex] a(b + c) = ab + ac [/latex]

Then, group the like terms and simplify using addition and subtraction.

Example 7.3-d: Multiplying Polynomials by Monomials

1. Multiply: [latex] 2x^3 [/latex] and [latex] (3x^2 + 2x - 5) [/latex]

   
   

2. Expand and simplify: [latex] 8x (x + 3) + 4x (x - 4) [/latex]

   
   

3. Expand and simplify: [latex] \displaystyle{\frac{1}{5}\{5y - 15[2 - 3(y - 2)] + 25\}} [/latex]

   
   

Solution

1. [latex] 2x^3 (3x^2 + 2x - 5) [/latex]   Expanding, by following the Product Rule of Exponents,

   [latex] = 6x^5 + 4x^4 - 10x^3 [/latex]

   
   

2. [latex] 8x (x + 3) + 4x (x - 4) [/latex]   Expanding,

   [latex] = 8x^2 + 24x + 4x^2 - 16x [/latex]   Grouping like terms,

   [latex] = 8x^2 + 4x^2 + 24x - 16x [/latex]   Adding and subtracting like terms,

   [latex] = 12x^2 + 8x [/latex]

   
   

3. [latex] \displaystyle{\frac{1}{5}\{5y - 15[2 - 3(y - 2)] + 25\}} [/latex]   Expanding the inner brackets,

   [latex] \displaystyle{= \frac{1}{5}\{5y - 15[2 - 3y + 6] + 25\}} [/latex]

   [latex] \displaystyle{= \frac{1}{5}\{5y - 30 + 45y - 90 + 25\}} [/latex]   Grouping like terms,

   [latex] \displaystyle{= \frac{1}{5}\{5y + 45y - 30 - 90 + 25\}} [/latex]   Adding and subtracting like terms,

   [latex] \displaystyle{= \frac{1}{5}\{50y - 95\}} [/latex]   Expanding the outer brackets,

   [latex] = 10y - 19 [/latex]

   
   

Multiplying a Binomial by a Binomial

When multiplying two binomials, each term of the first binomial is multiplied by each term of the second binomial. This is the same as adding the products of the First terms, Outside terms, Inside terms, and Last terms of each binomial, which can be remembered by the acronym "FOIL".

[latex] (a + b)(c + d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d [/latex]

The same result is obtained by using the distributive property to expand.

[latex] (a + b)(c + d) = a(c + d) + b(c + d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d [/latex]

Then, group the like terms and simplify using addition and subtraction.

Example 7.3-e: Multiplying Two Binomials

1. Multiply: [latex] (x + 5) [/latex] and [latex] (x + 6) [/latex]

   
   

2. Multiply: [latex] (2x + 3) [/latex] and [latex] (3x - 4) [/latex]

   
   

Solution

1. [latex] (x + 5)(x + 6) [/latex]

   [latex] = x^2 + 6x + 5x + 30 [/latex]

   [latex] = x^2 + 11x + 30 [/latex]

   
   

   or

   
   

   [latex] x(x + 6) + 5(x + 6) [/latex]

   [latex] = x^2 + 6x + 5x + 30 [/latex]

   [latex] = x^2 + 11x + 30 [/latex]

   
   

2. [latex] (2x + 3)(3x - 4) [/latex]

   [latex] = 6x^2 - 8x + 9x - 12 [/latex]

   [latex] = 6x^2 + x - 12 [/latex]

   
   

   or

   
   

   [latex] 2x(3x - 4) + 3(3x - 4) [/latex]

   [latex] = 6x^2 - 8x + 9x - 12 [/latex]

   [latex] = 6x^2 + x - 12 [/latex]

   
   

Special Products of Binomials

■ Squaring a binomial: the product of a binomial with itself

[latex] (a + b)^2 = (a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 [/latex]

[latex] (a - b)^2 = (a - b)(a - b) = a^2 - ab - ab + (-b)^2 = a^2 - 2ab + b^2 [/latex]

■ Difference of squares: the product of two binomials having the same two terms but opposite signs separating the terms; i.e., the product of the sum and difference of two terms.

[latex] (a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2 [/latex]

Example 7.3-f: Squaring a Binomial

Multiply the following expressions using the special product of binomials.

1. [latex] (2x + y)(2x + y) [/latex]

   
   

2. [latex] (3x + 4)(3x + 4) [/latex]

   
   

3. [latex] (3x - 2y)(3x - 2y) [/latex]

   
   

4. [latex] (5x - 6)(5x - 6) [/latex]

   
   

Solution

1. [latex] (2x + y)(2x + y) [/latex]

   [latex] = (2x + y)^2 [/latex]   Using [latex] (a + b)^2 = a^2 + 2ab + b^2 [/latex],

   [latex] = (2x)^2 + 2(2x)(y) + (y)^2 [/latex]

   [latex] = 4x^2 + 4xy + y^2 [/latex]

   
   

2. [latex] (3x + 4)(3x + 4) [/latex]

   [latex] = (3x + 4)2 [/latex]   Using [latex] (a + b)^2 = a^2 + 2ab + b^2 [/latex],

   [latex] = (3x)^2 + 2(3x)(4) + (4)^2 [/latex]

   [latex] = 9x^2 + 24x + 16 [/latex]

   
   

3. [latex] (3x - 2y)(3x - 2y) [/latex]

   [latex] = (3x - 2y)^2 [/latex]   Using [latex] (a - b)^2 = a^2 - 2ab + b^2 [/latex],

   [latex] = (3x)^2 - 2(3x)(2y) + (2y)^2 [/latex]

   [latex] = 9x^2 - 12xy + 4y^2 [/latex]

   
   

4. [latex] (5x - 6)(5x - 6) [/latex]

   [latex] = (5x - 6)^2 [/latex]   Using [latex] (a - b)^2 = a^2 - 2ab + b^2 [/latex],

   [latex] = (5x)^2 - 2(5x)(6) + (6)^2 [/latex]

   [latex] = 25x^2 - 60x + 36 [/latex]

   
   

Example 7.3-g: Multiplying the Sum and Difference of Two Terms (Difference of Squares)

Multiply the following expressions using the special product of binomials.

1. [latex] (3x + y)(3x - y) [/latex]

   
   

2. [latex] (2x + 5)(2x - 5) [/latex]

   
   

Solution

1. [latex] (3x + y)(3x - y) [/latex]   Using [latex] (a + b)(a - b) = a^2 - b^2 [/latex],

   [latex] = (3x)^2 - (y)^2 [/latex]

   [latex] = 9x^2 - y^2 [/latex]

   
   

2. [latex] (2x + 5)(2x - 5) [/latex]   Using [latex] (a + b)(a - b) = a^2 - b^2 [/latex],

   [latex] = (2x)^2 - (5)^2 [/latex]

   [latex] = 4x^2 - 25 [/latex]

   
   

Multiplying a Polynomial by a Polynomial

When multiplying a polynomial by a polynomial, multiply each term of the first polynomial by each term of the second polynomial. Then, using the distributive property, group the like terms and simplify using addition and subtraction.

Example 7.3-h: Multiplying Polynomials by Polynomials

1. Multiply: [latex] (x^2 + 7) [/latex] and [latex] (2x^2 + 5x + 2) [/latex]

   
   

2. Multiply: [latex] (x - 4) [/latex] and [latex] (2x^2 - x - 3) [/latex]

   
   

3. Expand and simplify: [latex] (x + 5)(2x - 6) + (3x - 4)(x - 5) [/latex]

   
   

4. Expand and simplify: [latex] (x - 3)(3x - 1) - (2x - 3)(x + 4) [/latex]

   
   

Solution

1. [latex] (x^2 + 7)(2x^2 + 5x + 2) [/latex]

   [latex] = x^2(2x^2 + 5x + 2) + 7(2x^2 + 5x + 2) [/latex]   Expanding,

   [latex] = 2x^4 + 5x^3 + 2x^2 + 14x^2 + 35x + 14 [/latex]   Adding like terms,

   [latex] = 2x^4 + 5x^3 + 16x^2 + 35x + 14 [/latex]

   
   

2. [latex] (x - 4) (2x^2 - x - 3) [/latex]

   [latex] = x(2x^2 - x - 3) - 4(2x^2 - x - 3) [/latex]   Expanding,

   [latex] = 2x3 - x 2 - 3x - 8x 2 + 4x + 12 [/latex]   Grouping like terms,

   [latex] = 2x^3 - x^2 - 8x^2 - 3x + 4x + 12 [/latex]   Adding and subtracting like terms,

   [latex] = 2x^3 - 9x^2 + x + 12 [/latex]

   
   

3. [latex] (x + 5)(2x - 6) + (3x - 4)(x - 5) [/latex]

   [latex] = [x(2x - 6) + 5(2x - 6)] + [3x(x - 5) - 4(x - 5)] [/latex]   Expanding,

   [latex] = (2x^2 - 6x + 10x - 30) + (3x^2 - 15x - 4x + 20) [/latex]   Removing the brackets,

   [latex] = 2x^2 - 6x + 10x - 30 + 3x^2 - 15x - 4x + 20 [/latex]   Grouping like terms,

   [latex] = 2x^2 + 3x^2 - 6x + 10x - 15x - 4x - 30 + 20 [/latex]   Adding and subtracting like terms,

   [latex] = 5x^2 - 15x - 10 [/latex]

   
   

4. [latex] (x - 3)(3x - 1) - (2x - 3)(x + 4) [/latex]

   [latex] = [x(3x - 1) - 3(3x - 1)] - [2x(x + 4) - 3(x + 4)] [/latex]   Expanding,

   [latex] = (3x^2 - x - 9x + 3) - (2x^2 + 8x - 3x - 12) [/latex]   Removing the brackets by distributing the negative sign,

   [latex] = 3x^2 - x - 9x + 3 - 2x^2 - 8x + 3x + 12 [/latex]   Grouping like terms,

   [latex] = 3x^2 - 2x^2 - x - 9x - 8x + 3x + 3 + 12 [/latex]   Adding and subtracting like terms,

   [latex] = x^2 - 15x + 15 [/latex]

   
   

Division

Dividing a Monomial by a Monomial

Dividing a monomial by another monomial is also just simplifying an algebraic term, as we did in the previous section. Divide the coefficients and divide the variables using the properties of exponents, where applicable.

Example 7.3-i: Dividing Monomials by Monomials

1. Divide [latex] 8x^2y [/latex] by [latex] 6x [/latex]

   
   

2. Divide [latex] -9x^2 [/latex] by [latex] 3x^3 [/latex]

   
   

Solution

1. [latex] \displaystyle{\frac{8x^2y}{6x} = \frac{8}{6} \cdot \frac{x^2}{x} \cdot y = \frac{4}{3}xy} [/latex] or [latex] \displaystyle{\frac{4xy}{3}} [/latex]

   
   

2. [latex] \displaystyle{\frac{-9x^2}{3x^3} = \frac{-9}{3} \cdot \frac{x^2}{x^3} = \frac{-3}{1} \cdot \frac{1}{x} = \frac{-3}{x}} [/latex]

   
   

Dividing a Polynomial by a Monomial

When dividing a polynomial by a monomial, divide each term of the polynomial by the monomial. The process is similar to dividing a monomial by a monomial.

Example 7.3-j: Dividing Polynomials by Monomials

1. Divide [latex] (9x^3 + 12x^2) [/latex] by [latex] 6x [/latex]

   
   

2. Divide [latex] (4x^4 + 2x^3 - 7x) [/latex] and [latex] 4x^4 [/latex]

   
   

Solution

1. [latex] \displaystyle{\frac{9x^3 + 12x^2}{6x} = \frac{9x^3}{6x} + \frac{12x^2}{6x} = \frac{3x^2}{2} + 2x} [/latex]

   
   

2. [latex] \displaystyle{\frac{4x^4 + 2x^3 - 7x}{4x^4} = \frac{4x^4}{4x^4} + \frac{2x^3}{4x^4} - \frac{7x}{4x^4} = 1 + \frac{1}{2x} - \frac{7}{4x^3}} [/latex]

   
   

7.3 Exercises

For Problems 1 to 8, simplify and evaluate the expressions.

1. [latex] 6y + 4y - 7y [/latex], where [latex] y = 10 [/latex]

   
   

2. [latex] 3x + 5x - 8x [/latex], where [latex] x = 4 [/latex]

   
   

3. [latex] 2z - z + 7z [/latex], where [latex] z = 7 [/latex]

   
   

4. [latex] 3A - A + 6A [/latex], where [latex] A = 10 [/latex]

   
   

5. [latex] (6x)(3x) - (5x)(4x) [/latex], where [latex] x = 3 [/latex]

   
   

6. [latex] (10x \times 4.5x) - (11x \times 4x) [/latex], where [latex] x = 50 [/latex]

   
   

7. [latex] (2x)(0.5x + 4x)(5x + x) [/latex], where [latex] x = 5 [/latex]

   
   

8. [latex] (4x)(12x + 0.25x)(0.5x + x) [/latex], where [latex] x = 3 [/latex]

   
   

For Problems 9 to 28, simplify the expressions.

9. [latex] 13x^2 + 8x - 2x^2 + 9x [/latex]

   
   

10. [latex] 7x + 12x^2 - 4x + 5x^2 [/latex]

    
    

11. [latex] -18y - 5y^2 + 19y - 2y^2 [/latex]

    
    

12. [latex] -14y - 2y^2 + 7y + 7y^2 [/latex]

    
    

13. [latex] 6x - 3x + 2y^2 + y^2 [/latex]

    
    

14. [latex] 9x^2 - 6x^2 + 7y - 6y [/latex]

    
    

15. [latex] 4xy^2 - x^2y^2 - 3xy^2 + 2x^2y^2 [/latex]

    
    

16. [latex] 3x^2y^2 - 2xy^2 - 8x^2y^2 + xy^2 [/latex]

    
    

17. [latex] 3[(5 - 3)(4 - x)] - 2 - 5[3(5x - 4) + 8] - 9x [/latex]

    
    

18. [latex] (5 - 14){x - 8[3 - 5(2x - 3) + 3x] - 3} [/latex]

    
    

19. [latex] 6[4(8 - y) - 5(3 + 3y)] - 21 - 7 [3(7 + 4y) - 4] + 198y [/latex]

    
    

20. [latex] \displaystyle{\frac{1}{2}\{y - 15[2 - 3(3y - 2) - 7y] -4\}} [/latex]

    
    

21. [latex] y - \{4x - [y - (2y - 9) - x] + 2\} [/latex]

    
    

22. [latex] 2y + \{-6y - [3x + (-4x + 3)] + 5\} [/latex]

    
    

23. [latex] (x - 1) - \{[x - (x - 3)] - x\} [/latex]

    
    

24. [latex] 9x - \{3y +[4x -(y - 6x)] - (x + 7y)\} [/latex]

    
    

25. [latex] 5\{-2y + 3[4x - 2(3 + x)]\} [/latex]

    
    

26. [latex] 4\{-7y + 8[5x - 3(4x + 6)]\} [/latex]

    
    

27. [latex] 2y + \{8[3(2y - 5) - (8y + 9) + 6]\} [/latex]

    
    

28. [latex] 7x - \{5[4(3x - 8) - (9x + 10)] + 14\} [/latex]

    
    

For Problems 29 to 38, expand and simplify the expressions.

29. [latex] (2y - 1)(y - 4) - (3y + 2)(3y - 1) [/latex]

    
    

30. [latex] (y + 4)(y - 3) + (y - 2)(y - 3) [/latex]

    
    

31. [latex] (2x + 3)(2x - 1) - 4(x^2 - 7) [/latex]

    
    

32. [latex] 4(2x - 1)(x + 3) - 3(x - 2)(3x - 4) [/latex]

    
    

33. [latex] 3(x - 2)(4 - 3x) + 4(2x - 1)(3 - x) [/latex]

    
    

34. [latex] 2(3x + 2)(1 - 3x) + 3(2x - 1)(4 - x) [/latex]

    
    

35. [latex] 4(3x^2 + 4) - 2(x + 3)(x + 5) [/latex]

    
    

36. [latex] 3(5x^2 - 1) - (2x - 4)(3x + 5) [/latex]

    
    

37. [latex] 3(2 - 3x)(2 + x) - (1 - x)(x - 3) [/latex]

    
    

38. [latex] (x - 2)(3x + 2) - (3x + 2)(x - 5) [/latex]

    
    

For Problems 39 to 54, expand the expressions by using special products of binomials.

39. [latex] (x + 5)^2 [/latex]

    
    

40. [latex] (x + 7)^2 [/latex]

    
    

41. [latex] (2x + 3y)^2 [/latex]

    
    

42. [latex] (3x + 4y)^2 [/latex]

    
    

43. [latex] (3 - x)^2 [/latex]

    
    

44. [latex] (7 - x)^2 [/latex]

    
    

45. [latex] (3x - 2y)^2 [/latex]

    
    

46. [latex] (2x - 3y)^2 [/latex]

    
    

47. [latex] (1 - 3x)^2 [/latex]

    
    

48. [latex] (6x - 1)^2 [/latex]

    
    

49. [latex] (3 - 2x)^2 [/latex]

    
    

50. [latex] (2y - 1)^2 [/latex]

    
    

51. [latex] (x + 5)(x - 5) [/latex]

    
    

52. [latex] (12x - 1)(12x + 1) [/latex]

    
    

53. [latex] (3 + 7x)(3 - 7x) [/latex]

    
    

54. [latex] (2a + 9b)(2a - 9b) [/latex]

    
    

For Problems 55 to 62, expand and simplify the expressions.

55. [latex] (x + 3)^2 + (x - 2)^2 [/latex]

    
    

56. [latex] (x + 5)^2 + (x - 4)^2 [/latex]

    
    

57. [latex] (4 + x)^2 - (x - 3)(x + 3) [/latex]

    
    

58. [latex] (3 + x)^2 + (x + 5)(x - 5) [/latex]

    
    

59. [latex] (3x - 2)^2 + (2x - 3)(2x + 3) [/latex]

    
    

60. [latex] (2x + 5)(2x - 5) + (1 - 4x)^2 [/latex]

    
    

61. [latex] (2x - 4)^2 - (y + 3)^2 [/latex]

    
    

62. [latex] (5x - 6)^2 - (x + 5)^2 [/latex]

    
    

For Problems 63 to 72, simplify the expressions.

63. [latex] \displaystyle{\frac{(16y)(8x)}{(4x)(8y)}} [/latex]

    
    

64. [latex] \displaystyle{\frac{(20y)(4x)}{(2x)(5y)}} [/latex]

    
    

65. [latex] \displaystyle{\frac{(6x)(-18y)}{(3x)(-24y)}} [/latex]

    
    

66. [latex] \displaystyle{\frac{(7x)(18y)}{(14x)(-27y)}} [/latex]

    
    

67. [latex] \displaystyle{\frac{-x^2y - xy^2}{xy}} [/latex]

    
    

68. [latex] \displaystyle{\frac{x^2y - 3xy^2}{xy}} [/latex]

    
    

69. [latex] \displaystyle{\frac{x^2y - 3xy^2 + 4x^2y + xy}{xy}} [/latex]

    
    

70. [latex] \displaystyle{\frac{3x^3y^3 - 6x^2y + 3xy^2 + 3xy}{3xy}} [/latex]

    
    

71. [latex] \displaystyle{\frac{6xy^2}{7} \cdot \frac{21x^2}{y} \cdot \frac{1}{36xy^2}} [/latex]

    
    

72. [latex] \displaystyle{\frac{12x^2y^3}{5} \cdot \frac{15x^2}{4xy} \cdot \frac{1}{30x^3y}} [/latex]