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Study Guides > College Algebra

Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:
(a+2b)(4abc)a(4abc)+2b(4abc)Use the distributive property.4a2abac+8ab2b22bcMultiply.4a2+(ab+8ab)ac2b22bcCombine like terms.4a2+7abac2bc2b2Simplify.\begin{array}{cc}\left(a+2b\right)\left(4a-b-c\right)\hfill & \hfill \\ a\left(4a-b-c\right)+2b\left(4a-b-c\right)\hfill & \text{Use the distributive property}.\hfill \\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\hfill & \text{Multiply}.\hfill \\ 4{a}^{2}+\left(-ab+8ab\right)-ac - 2{b}^{2}-2bc\hfill & \text{Combine like terms}.\hfill \\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\hfill & \text{Simplify}.\hfill \end{array}

Example 8: Multiplying Polynomials Containing Several Variables

Multiply (x+4)(3x2y+5)\left(x+4\right)\left(3x - 2y+5\right).

Solution

Follow the same steps that we used to multiply polynomials containing only one variable.
x(3x2y+5)+4(3x2y+5)Use the distributive property.3x22xy+5x+12x8y+20Multiply.3x22xy+(5x+12x)8y+20Combine like terms.3x22xy+17x8y+20Simplify.\begin{array}{cc}x\left(3x - 2y+5\right)+4\left(3x - 2y+5\right) \hfill & \text{Use the distributive property}.\hfill \\ 3{x}^{2}-2xy+5x+12x - 8y+20\hfill & \text{Multiply}.\hfill \\ 3{x}^{2}-2xy+\left(5x+12x\right)-8y+20\hfill & \text{Combine like terms}.\hfill \\ 3{x}^{2}-2xy+17x - 8y+20 \hfill & \text{Simplify}.\hfill \end{array}

Try It 8

(3x1)(2x+7y9)\left(3x - 1\right)\left(2x+7y - 9\right). Solution

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  • College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.