Expanding and Simplifying Expressions

Expanding and simplifying is something we do in algebra all the time. It is such an important skill. Here are some examples of expanding and simplifying expressions, followed by an exercise containing 24 questions to try yourself.

Example 1

\(\begin{array}{l} 4+2(x-3) \\ =4+2x-6 \\ =2x-2 \end{array}\)

Example 2

\(\begin{array}{l}3(2x+1)-4(x-3) \\ =6x+3-4x+12 \\ =2x+15\end{array}\)

In example 2 we have used the fact \((-4) \times (-3)=+12\)

Example 3

\(\begin{array}{l}3(2x-3y)+4(y-x) \\ =6x-9y+4y-4x \\ =2x-5y\end{array}\)

Example 4

\(\begin{array}{l}(x+2y)(2x-y) \\ =2x^2-xy+4xy-2y^2 \\ =2x^2+3xy-2y^2\end{array}\)

Example 5

\(\begin{array}{l}(x+y)^2 \\ =(x+y)(x+y) \\ =x^2+xy+yx+y^2 \\ =x^2+2xy+y^2\end{array}\)

Example 6

\(\begin{array}{l}(x-y)^2 \\ =(x-y)(x-y) \\ =x^2-xy-yx+y^2 \\ =x^2-2xy+y^2\end{array}\)

Example 7

\(\begin{array}{l}(x+y)(x-y) \\ =x^2-xy+yx-y^2 \\ =x^2-y^2\end{array}\)

Example 7 is known as the Difference of Two Squares (D.O.T.S.) (i.e. x squared take away y squared). It is very important.

You should make an effort to learn the following expansions so you can easily recognise their pattern in the future. They will be very useful when practising factorising:

\[\begin{array}{rcl}(x+y)^2 & \equiv & x^2+2xy+y^2\\
(x-y)^2 & \equiv & x^2-2xy+y^2 \\
(x+y)(x-y) & \equiv & x^2-y^2\end{array}\]

The three lines \(\equiv\) here means ‘identically equal to’ and are used when an ‘equation’ is true for all values of x and y.

If there are three (or more) brackets to multiply together, you simply do two of them, tidy that up and continue along until they are all done, as shown in the examples below:

Example 8

\(\begin{array}{l}2(x-2)(2x+1) \\ =(2x-4)(2x+1) \\ =4x^2+2x-8x-4 \\ =4x^2-6x-4\end{array}\)

Example 9

\(\begin{array}{l}(x-2)(x+1)(2x+3) \\ =(x-2)(2x^2+3x+2x+3) \\ =(x-2)(2x^2+5x+3) \\ =2x^3+5x^2+3x-4x^2-10x-6 \\ =2x^3+x^2-7x-6\end{array}\)


Exercise

Expand and collect terms in the following:

  1. \((x+y)(x-2y)\)
  2. \(2x+3(x+1)\)
  3. \(2(2x-3)-3(x-1)\)
  4. \(2(x+2y)+3(y-2x)\)
  5. \(2(2x+y)(x-y)\)
  6. \((2x+3)(3x-2)\)
  7. \((2x+1)(x+2)\)
  8. \((x+2)(x-2)\)
  9. \(2(x+3)(x-1)\)
  10. \((x+1)(2x+1)(x-1)\)
  11. \(4(3-2x)-(x-7)\)
  12. \((2x-1)(2x+1)\)
  13. \((2x+3)(2x-3)\)
  14. \((5-x)(5+x)\)
  15. \((x+7)^2\)
  16. \((x+1)^3\)
  17. \((x+1)(x-1)(x+2)\)
  18. \((4x+1)(x+2)\)
  19. \((3x-1)(3x-1)\)
  20. \((2x+1)(x-3)(x+2)\)
  21. \((x+3)(x-2)(1-x)\)
  22. \((1-x)(1+x)\)
  23. \(4(2+x)(3-x)\)
  24. \(5(2x+1)^2\)