Algebra

NZ Level 7 (NZC) Level 2 (NCEA)

Simplify and manipulate algebraic expressions

Lesson

We've already learnt that there are a number of conventions (rules) which need to be followed in order to solve problems with different operations correctly. This is known as the order of operations.

Remember!

The order goes:

Step 1: Do operations inside grouping symbols such as parentheses (...), brackets [...] and braces {...}.

Step 2: Do multiplication (including powers) and division (including roots) going from left to right.

Step 3: Do addition and subtraction going from left to right.

Each time we complete one of these steps, we simplify our equation until we get one final answer.

Let's look at an example so we can see this process in action.

Simplify: $6x^2+4x^2\times5+7x^2$6`x`2+4`x`2×5+7`x`2

Think: Multiplication comes before addition. Then collect like terms.

Do: | $6x^2+4x^2\times5+7x^2$6x2+4x2×5+7x2 |
$=$= | $6x^2+20x^2+7x^2$6x2+20x2+7x2 |

$=$= | $33x^2$33x2 |

Many questions that you will encounter will often require **expanding**. As revision, we've summarised some methods of expansion in the table below.

Factorised form | Expanded form |
---|---|

$A\left(B+C\right)$A(B+C) |
$AB+AC$AB+AC |

$A\left(B+C+D\right)$A(B+C+D) |
$AB+AC+AD$AB+AC+AD |

$\left(A+B\right)\left(C+D\right)$(A+B)(C+D) |
$AC+AD+BC+BD$AC+AD+BC+BD |

$\left(A+B\right)^2$(A+B)2 |
$A^2+2AB+B^2$A2+2AB+B2 |

$\left(A-B\right)^2$(A−B)2 |
$A^2-2AB+B^2$A2−2AB+B2 |

$\left(A+B\right)\left(A-B\right)$(A+B)(A−B) |
$A^2-B^2$A2−B2 |

Let's look at a few examples.

Expand and simplify: $\left(5x-9\right)\left(5x+9\right)$(5`x`−9)(5`x`+9)

Think: This expression is of the form $\left(A-B\right)\left(A+B\right)$(`A`−`B`)(`A`+`B`), so its expanded form will be a difference of two squares.

Do: | $\left(5x-9\right)\left(5x+9\right)$(5x−9)(5x+9) |
$=$= | $\left(5x\right)^2-9^2$(5x)2−92 |

$=$= | $25x^2-81$25x2−81 |

Expand and simplify: $\left(\frac{6y}{x}-\frac{3x}{y}\right)^2$(6`y``x`−3`x``y`)2

Think: This expression is of the form $\left(A-B\right)^2$(`A`−`B`)2, so its expanded form will look like $A^2-2AB+B^2$`A`2−2`A``B`+`B`2.

Do: | $\left(\frac{6y}{x}-\frac{3x}{y}\right)^2$(6yx−3xy)2 |
$=$= | $\left(\frac{6y}{x}\right)^2-2\frac{6y}{x}\frac{3x}{y}+\left(\frac{3x}{y}\right)^2$(6yx)2−26yx3xy+(3xy)2 |

$=$= | $\frac{36y^2}{x^2}-36+\frac{9x^2}{y^2}$36y2x2−36+9x2y2 |

Try simplifying the following algebraic expressions below.

Expand and simplify:

$6\left(10x+8y\right)-6\left(-6y-6x\right)$6(10`x`+8`y`)−6(−6`y`−6`x`)

Simplify $\frac{8x^2}{11}+\frac{5x^2-2x}{55}$8`x`211+5`x`2−2`x`55.

Simplify the following expression:

$\frac{x+3}{x+4}\div\frac{x+3}{3}-\frac{1}{x+4}$`x`+3`x`+4÷`x`+33−1`x`+4

Expand and simplify: $\left(4st+\frac{2}{st}-4\right)\left(4st+\frac{2}{st}+4\right)$(4`s``t`+2`s``t`−4)(4`s``t`+2`s``t`+4)

Manipulate rational, exponential, and logarithmic algebraic expressions

Apply algebraic methods in solving problems