7.2 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 7.2-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 the 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 the 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 7.2-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. Where applicable, multiply the coefficients.

Evaluate the following expressions:

  1. Multiply [latex]6x^2y[/latex] and [latex]5[/latex]
  2. Multiply [latex](3a^3)[/latex], [latex](-4)[/latex] and [latex](2)[/latex]

Solution

  1. [latex](6x^2y)(5)[/latex][latex]= (6)(5)(x^2)(y)[/latex][latex]= 30x^2y[/latex]
  2. [latex](3a^3)(-4)(2)[/latex][latex]= (3)(-4)(2)(a^3)[/latex][latex]= -24a^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 7.2-d: Multiplying Polynomials by Monomials

  1. Multiply: [latex]2[/latex] and [latex](3x^2 + 2x - 5)[/latex]
  2. Expand and simplify: [latex]8 (x + 3) + 4 (x - 4)[/latex]

Solution

  1. [latex]2 (3x^2 + 2x - 5)[/latex]
    Expanding, by following the Distributive Property, [latex]= 6x^2 + 4x - 10[/latex]
  2. [latex]8 (x + 3) + 4 (x - 4)[/latex]
    Expanding,[latex]= 8x + 24 + 4x - 16[/latex]
    Grouping like terms, [latex]= 8x + 4x + 24 - 16[/latex]
    Adding and subtracting like terms,[latex]= 12x + 8[/latex]

Division

Dividing a Monomial by a Monomial

Dividing a monomial by another monomial simplifies an algebraic term, as we did in the previous section. Divide the coefficients, where applicable.

Example 7.2-i: Dividing Monomials by Monomials

  1. Divide [latex]8x^2y[/latex] by [latex]6[/latex]
  2. Divide [latex]-9x^2[/latex] by [latex]3[/latex]

Solution

  1. [latex]\displaystyle{\frac{8x^2y}{6}}[/latex] = [latex]\displaystyle{\frac{4x^2y}{3}}[/latex]
  2. [latex]\displaystyle{\frac{-9x^2}{3}}[/latex] = [latex]- 3x^2[/latex]

7.2 Exercises

Answers to the odd-numbered problems are available at the end of the textbook.

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.

  1. [latex]13x^2 + 8x - 2x^2 + 9x[/latex]
  2. [latex]7x + 12x^2 - 4x + 5x^2[/latex]
  3. [latex]-18y - 5y^2 + 19y - 2y^2[/latex]
  4. [latex]-14y - 2y^2 + 7y + 7y^2[/latex]
  5. [latex]6x - 3x + 2y^2 + y^2[/latex]
  6. [latex]9x^2 - 6x^2 + 7y - 6y[/latex]
  7. [latex]4xy^2 - x^2y^2 - 3xy^2 + 2x^2y^2[/latex]
  8. [latex]3x^2y^2 - 2xy^2 - 8x^2y^2 + xy^2[/latex]
  9. [latex]3[(5 - 3)(4 - x)] - 2 - 5[3(5x - 4) + 8] - 9x[/latex]
  10. [latex](5 - 14){x - 8[3 - 5(2x - 3) + 3x] - 3}[/latex]
  11. [latex]6[4(8 - y) - 5(3 + 3y)] - 21 - 7 [3(7 + 4y) - 4] + 198y[/latex]
  12. [latex]\displaystyle{\frac{1}{2}\{y - 15[2 - 3(3y - 2) - 7y] -4\}}[/latex]
  13. [latex]y - \{4x - [y - (2y - 9) - x] + 2\}[/latex]
  14. [latex]2y + \{-6y - [3x + (-4x + 3)] + 5\}[/latex]
  15. [latex](x - 1) - \{[x - (x - 3)] - x\}[/latex]
  16. [latex]9x - \{3y +[4x -(y - 6x)] - (x + 7y)\}[/latex]
  17. [latex]5\{-2y + 3[4x - 2(3 + x)]\}[/latex]
  18. [latex]4\{-7y + 8[5x - 3(4x + 6)]\}[/latex]
  19. [latex]2y + \{8[3(2y - 5) - (8y + 9) + 6]\}[/latex]
  20. [latex]7x - \{5[4(3x - 8) - (9x + 10)] + 14\}[/latex]

Unless otherwise indicated, this chapter is an adaptation of the eTextbook Foundations of Mathematics (3rd ed.) by Thambyrajah Kugathasan, published by Vretta-Lyryx Inc., with permission. Adaptations include supplementing existing material and reordering chapters.

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