Algebraic Manipulation, Expanding and Factorising
Simplifying, factorising, cancelling and rearranging all change how an expression looks without changing what it is worth, so one test number can check every one of them.
Get the method right under pressure
Free interactive practice on the steps that lose marks under exam pressure.
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Same value, new shape
Almost every algebra question asks you to rewrite something: simplify it, factorise it, cancel it or rearrange it. Each of those moves changes how an expression LOOKS without changing what it is WORTH, and that gives you one check that works on all of them. Pick a number, put it into the expression you started with and into the one you finished with, and compare the two results. If they disagree, a mistake has been made somewhere, even before you know where. This module covers the moves beyond simple brackets: multiplying and dividing terms using the rules for powers, factorising fully, factorising a quadratic with a number in front of x squared, simplifying a fraction with letters in it, and changing the subject of a formula, including when the new subject turns up in more than one place. It keeps coming back to that single check, and to the reason it works, because the check is what turns a guess into an answer you can trust.
Five words for rewriting algebra
Two of these describe the parts of an expression, two describe kinds of statement, and one describes what a formula is arranged to find.
Match each algebra task to its first move
- Simplify 3a times 4a squared
- Factorise 6x squared + 9x
- Simplify (x squared - 9) over (x + 3)
- Make t the subject of v = u + at
- Work out 2x squared when x = -3
- Multiply the numbers, then combine the letters by adding their powers
- Take out the highest factor shared by both terms, including a letter they share
- Factorise the top first, then remove a whole bracket that appears above and below
- Undo what has been done to the letter, in reverse order, on both sides
- Square the number before doubling it, remembering a negative number squared is positive
Multiply two algebraic terms
Simplify 3x squared times 4x cubed.
- 12x to the power 5
- 12x to the power 6
- 7x to the power 5
- 7x to the power 6
Factors can cancel, terms cannot
A fraction with letters in it simplifies exactly as a number fraction does: divide the top and the bottom by the same thing. The whole question is what counts as the same thing, and the answer is a factor, never a term.
Simplify a fraction with a reason
Choose the option in each gap to simplify (2x + 8) over 2 and justify it.
Spot the two broken simplifications
Six statements about simplifying and factorising. Tap the TWO that are wrong.
- (2x + 6) over 2 simplifies to x + 6.
- x to the power 5 divided by x squared is x cubed.
- 4x squared - 8x factorised fully is 4(x squared - 2x).
- (3x) squared is the same as 9x squared.
- (x squared + 4x) over x simplifies to x + 4, for any x that is not zero.
- 10ab over 5b simplifies to 2a.
Two methods that need a plan
Two of the Higher moves go wrong when they are attempted by guesswork, and both have a reliable plan. To factorise 3x squared + 10x + 8, where x squared has a number in front, multiply that number by the last one: 3 times 8 is 24. Find two numbers that multiply to 24 and add to 10, which are 4 and 6, and use them to split the middle: 3x squared + 4x + 6x + 8. Factorise the pairs, x(3x + 4) + 2(3x + 4), and the shared bracket gives (3x + 4)(x + 2). A check with x = 1 gives 21 both ways. To change the subject when the new subject appears in two terms, say kx + 3 = 2x + m, gather every term with x on one side and everything else on the other: kx - 2x = m - 3. Now x is a common factor, so take it out, x(k - 2) = m - 3, and divide by the bracket. The step people miss is that factorising, which here makes x appear once.
Factorise with a number before x squared
2x squared + 11x + 15 can be written as (2x + a)(x + b), where a and b are whole numbers. What is a?
Rearrange when the subject appears twice
The letter you want as the new subject appears in two terms. Put the five steps in order.
- Clear any fractions or brackets so that every term sits on one line
- Gather the terms containing the new subject on one side and all the others on the other side
- Take the new subject outside a bracket as a common factor
- Divide both sides by that bracket
- Check by putting one set of numbers into the old and the new formula
Sending a story there and back
A publisher has a short story translated into Welsh. Before it is printed, an editor hands the Welsh version to a second translator who has never seen the original, and asks for it to be put back into English. Then the two English versions are laid side by side. Nobody needs the second translator to choose the same words as the author did; the only question is whether the meaning came back unchanged. If one sentence returns saying something different, that is where the first translation slipped, even though each word in it looked reasonable on its own. A change of wording is safe only when it keeps the meaning, and the quickest way to find out is to make the round trip and compare what comes back with what was sent.
Complete the checking rules
A statement that holds for _____ value of the letter is an identity. In an algebraic fraction, only a whole _____ of the top and bottom can be cancelled, so _____ first. To check any rearrangement, put the same _____ into the old and new versions and compare.
Which are true for every x?
Three of these hold whatever number x stands for. Choose all three.
- 3(x + 2) = 3x + 6
- (x + 3) squared = x squared + 9
- x squared - 16 = (x + 4)(x - 4)
- 2x + 5 = 11
- (2x + 1)(x + 3) = 2x squared + 7x + 3
- 6x over 2x = 3x
The manipulation run
Five quick questions on substituting, simplifying, factorising and rearranging. Three lives.
Check a friend without redoing it
A friend has asked you to check their algebra homework, and you do not have time to redo every question. Work through three decisions.
- Their first answer says (x squared + 5x) over x simplifies to x squared + 5. You want to know whether it is right as quickly as possible. What do you do?
- Next, they made w the subject of P = 2l + 2w and wrote w = P - l. You know a rectangle 3 long and 4 wide has a perimeter of 14. What do you decide?
- Finally, they claim (x + 1) squared = x squared + 1 is an identity. They checked it with x = 0 and got 1 on both sides. What do you conclude?
Rearrange with the subject in two terms
A friend was asked to make x the subject of y = (x + 4) over (x - 2). They multiplied both sides by (x - 2) to get yx - 2y = x + 4, then stopped and wrote that it cannot be done because x is on both sides. Explain why it can be done, finish the rearrangement, and check your answer with a test number.
- Why x appearing in two terms is not a dead end
- The line with every x term on one side
- The bracket that x is taken out of
- The final formula for x
- A test number that confirms it