Simplifying & Manipulating Expressions

Collecting like terms

Collecting like terms involves ‘putting’ expressions with the same terms together. For the higher paper you need to be able to do this with decimals, fractions and surds. This is a key skill that you will also need when you expand brackets, so make sure you are confident and double check your work! What makes terms the same? Terms that are the same have the same variable. For example, the term a is different from the term a2, whereas the term 3b is the same as the term 12b. And remember that you can only collect surds that have the same number underneath.

A foolproof way to collect like terms.

Step 1: Write out the expression.

4t⁵ - 3g + ½ t - 6t⁵ + 5t + 6g

Step 2: Colour code or draw different shapes around different terms. Make sure you include the sign - the sign is always to the left of the number!!!!

4t⁵ - 3g + ½ t -6t⁵ + 5t + 6g

Step 3: Add or subtract the same terms together:

=4t⁵ - 3g + ½ t -6t⁵ + 5t +6g

=4t⁵ - 6t⁵ +6g -3g + ½ t + 5t

=-2t⁵ + 3g + 5½t

You can apply this method to more complicated questions such as:

=√__5 + 9g - 6√__5 + 0.3t - ⅓ g

=5 - 6__√__5 + 0.3t - ⅓ g + 9g

=-5__√__5 + 0.3t - 9 ⅓ g

Taking out common factors

When you have expressions, sometimes you will be able to simplify them. To do this you need to take out a common factor of the expression.

Remember, a factor is a number or a term that can be divided without a remainder into your variables. Eg. 14y + 21a

If we look at the factors of each of the terms we see that there are some common factors.

14y has the factors: 1, 14, y, 2,7. We need to remember that one of the factors will include the y variable.

21a has the factors: 1, 21, a, 3,7. We need to remember that one of the factors will include the y variable.

If we compare the two lists of factors we can see that they both have a 7. Therefore we can divide our whole expression by a factor of 7.

So 14y + 21a, divided by a factor of 7, will be 2y + 3a.

To write this algebraically, we use brackets:

14y + 21a = 7(2y +3a). This is our answer.

Eg 2. Factorise __10t + 15t²

Once again we need to look at the factors of each number: 10t x__t __we also need to multiply by t!

1 x 10

2 x 5

15t² In this case we have t² we need to remember that this means t x t

1 x 15

3 x 5

To factorise this expression we can take out a factor of 5 and t, because in both expression we multiply by t. Therefore, when factorised, this expression is __5t __(2 +2t)

Check your answer by expanding your bracket! If you don’t get the expression you started with then you have made a mistake!

Expand the brackets: 4(a +5) -2(3a +6)

Your answer should include: 8 - 2a / -2a + 8