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)(4a−b−c)a(4a−b−c)+2b(4a−b−c)Use the distributive property.4a2−ab−ac+8ab−2b2−2bcMultiply.4a2+(−ab+8ab)−ac−2b2−2bcCombine like terms.4a2+7ab−ac−2bc−2b2Simplify.\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}(a+2b)(4a−b−c)a(4a−b−c)+2b(4a−b−c)4a2−ab−ac+8ab−2b2−2bc4a2+(−ab+8ab)−ac−2b2−2bc4a2+7ab−ac−2bc−2b2Use the distributive property.Multiply.Combine like terms.Simplify. Example 8: Multiplying Polynomials Containing Several Variables Multiply (x+4)(3x−2y+5)\left(x+4\right)\left(3x - 2y+5\right)(x+4)(3x−2y+5). Solution Follow the same steps that we used to multiply polynomials containing only one variable. x(3x−2y+5)+4(3x−2y+5)Use the distributive property.3x2−2xy+5x+12x−8y+20Multiply.3x2−2xy+(5x+12x)−8y+20Combine like terms.3x2−2xy+17x−8y+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}x(3x−2y+5)+4(3x−2y+5)3x2−2xy+5x+12x−8y+203x2−2xy+(5x+12x)−8y+203x2−2xy+17x−8y+20Use the distributive property.Multiply.Combine like terms.Simplify. Try It 8 (3x−1)(2x+7y−9)\left(3x - 1\right)\left(2x+7y - 9\right)(3x−1)(2x+7y−9). Solution Licenses & AttributionsCC licensed content, Specific attributionCollege 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.