 ## How to simplify Variable Expressions

Expressions are defined as an algebraic statement which contains several terms which are separated by mathematical operations. Now, these terms could be pure variable terms, pure constant terms, or even mixed terms such as variables with coefficients.

### Simplifying Variable Expressions

To simplify variable expressions, we must use 2 techniques:

• Firstly, we must add or subtract like terms like those with the same variable (like $$3x, \ -7x$$) or those with the same powers (like $$2x^2, \ -3x^2$$)
• Next, we must apply distributive law if possible. The distributive property states that multiplication distributes over addition, i.e., $$x(y \ + \ z) \ = \ (xy \ + \ xz)$$.

Example 1: $$2x^2(7x \ + \ 9) \ + \ x^2 \ = \ 2x^2(7x) \ + \ 2x^2(9) \ + \ x^2 = \ 14x^3 \ + \ 18x^2 \ + \ x^2 \ = \ 14x^3 \ + \ 19x^2$$

Example 2: $$7x^2(3x \ + \ 5) \ + \ x^2 \ = \ 7x^2(3x) \ + \ 7x^2(5) \ + \ x^2 \ = \ 21x^3 \ + \ 35x^2 \ + x^2 \ = \ 21x^3 \ + \ 36x^2$$

Example 3: $$x^3(x \ + \ 4) \ + \ 8x^2 \ = \ x^3(x) \ + \ x^3(4) \ + \ 8x^2 \ = \ x^4 \ + \ 4x^3 \ + \ 8x^2$$

Example 4: $$x^2(x \ - \ 7) \ + \ 2x^2 \ = \ x^2(x) \ + \ x^2(-7) \ + \ 2x^2 \ = \ x^3 \ - \ 7x^2 \ + \ 2x^2 \ = \ x^3 \ - \ 5x^2$$

Example 5: $$x^2(x \ - \ 3) \ + \ x^2 (x \ + \ 8) \ = \ x^2(x) \ + \ x^2(-3) \ + \ x^2(x) \ + \ x^2(8) \ =$$ $$x^3 \ - \ 3x^2 \ + \ x^3 \ + \ 8x^2 \ = \ 2x^3 \ + \ 5x^2$$

Example 6: $$2x^2(x \ + \ 6) \ - \ x^2(x \ + \ 7) \ = \ 2x^2(x) \ + \ 2x^2(6) \ - \ x^2(x) \ - \ x^2(7) \ =$$ $$2x^3 \ + \ 12x^2 \ - \ x^3 \ - \ 7x^2 \ = \ x^3 \ + \ 5x^2$$

Example 7: $$x^3(x \ - \ 4) \ + \ x^3(x \ - \ 8) \ = \ x^3(x) \ + \ x^3(-4) \ + \ x^3(x) + \ x^3(-8) \ =$$ $$x^4 \ - \ 4x^3 \ + \ x^4 \ - \ 8x^3 \ = \ 2x^4 \ - \ 12x^3$$

### Exercises for Simplifying Variable Expressions

1) $$19x^2 \ (7x \ + \ x) \ + \ x^3 \ =$$

2) $$18x^2 \ (7x \ + \ x) \ + \ x^3 \ =$$

3) $$17x^2 \ (6x \ + \ x) \ + \ x^3 \ =$$

4) $$16x^2 \ (2x \ + \ x) \ + \ x^3 \ =$$

5) $$15x^2 \ (4x \ + \ x) \ + \ x^3 \ =$$

6) $$18x^2 \ (5x) \ =$$

7) $$20x^2 \ (5x \ + \ x) \ =$$

8) $$12x^2 \ (2x \ + \ x) \ =$$

9) $$5x^2 \ (7x \ + \ x) \ =$$

10) $$3x^2 \ (3x) \ =$$

1) $$19x^2 \ (7x \ + \ x) \ + \ x^3 \ = \color{red}{153x^3}$$
2) $$18x^2 \ (7x \ + \ x) \ + \ x^3 \ = \color{red}{145x^3}$$
3) $$17x^2 \ (6x \ + \ x) \ + \ x^3 \ = \color{red}{120x^3}$$
4) $$16x^2 \ (2x \ + \ x) \ + \ x^3 \ = \color{red}{49x^3}$$
5) $$15x^2 \ (4x \ + \ x) \ + \ x^3 \ = \color{red}{76x^3}$$
6) $$18x^2 \ (5x) \ = \color{red}{90x^3}$$
7) $$20x^2 \ (5x \ + \ x) \ = \color{red}{120x^3}$$
8) $$12x^2 \ (2x \ + \ x) \ = \color{red}{36x^3}$$
9) $$5x^2 \ (7x \ + \ x) \ = \color{red}{40x^3}$$
10) $$3x^2 \ (3x) \ = \color{red}{9x^3}$$

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