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:

\begin{array}{cc} (a + 2 b) (4 a - b - c) \\ a (4 a - b - c) + 2 b (4 a - b - c) & Use the distributive property . \\ 4 a^{2} - a b - a c + 8 a b - 2 b^{2} - 2 b c & Multiply . \\ 4 a^{2} + (- a b + 8 a b) - a c - 2 b^{2} - 2 b c & Combine like terms . \\ 4 a^{2} + 7 a b - a c - 2 b c - 2 b^{2} & Simplify . \end{array}

$\begin{array}{cc}\left(a+2b\right)\left(4a-b-c\right)\hfill & \hfill \\ a\left(4a-b-c\right)+2b\left(4a-b-c\right)\hfill & \text{Use the distributive property}.\hfill \\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\hfill & \text{Multiply}.\hfill \\ 4{a}^{2}+\left(-ab+8ab\right)-ac - 2{b}^{2}-2bc\hfill & \text{Combine like terms}.\hfill \\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\hfill & \text{Simplify}.\hfill \end{array}$

Example 8: Multiplying Polynomials Containing Several Variables

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

Solution

Follow the same steps that we used to multiply polynomials containing only one variable.

\begin{array}{cc} x (3 x - 2 y + 5) + 4 (3 x - 2 y + 5) & Use the distributive property . \\ 3 x^{2} - 2 x y + 5 x + 12 x - 8 y + 20 & Multiply . \\ 3 x^{2} - 2 x y + (5 x + 12 x) - 8 y + 20 & Combine like terms . \\ 3 x^{2} - 2 x y + 17 x - 8 y + 20 & Simplify . \end{array}

$\begin{array}{cc}x\left(3x - 2y+5\right)+4\left(3x - 2y+5\right) \hfill & \text{Use the distributive property}.\hfill \\ 3{x}^{2}-2xy+5x+12x - 8y+20\hfill & \text{Multiply}.\hfill \\ 3{x}^{2}-2xy+\left(5x+12x\right)-8y+20\hfill & \text{Combine like terms}.\hfill \\ 3{x}^{2}-2xy+17x - 8y+20 \hfill & \text{Simplify}.\hfill \end{array}$

Try It 8

$\left(3x - 1\right)\left(2x+7y - 9\right)$ .

Solution