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This page sets out the method for expanding & factorising in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Expanding & factorising topic page.
When you see brackets with a number or letter in front, multiply out each bracket then collect like terms.
When you see two brackets multiplied together, use a 2 by 2 grid so no product is missed.
When you are asked to factorise and there is no with a number-only constant, look for what every term shares.
3(x + 2) = 3 × x + 3 × 2 = 3x + 6.
Every term inside the bracket is multiplied by the factor outside.
A single bracket is this topic. Two brackets, such as (x + 2)(x + 3), come with quadratics.
Do not stop after multiplying only the first term.
3x + 6 = 3(x + 2).
The factor outside is the highest common factor of the terms.
Expand your factorised answer. You should get the expression you started with.
If a term is missing, the factor was not multiplied by every term.
4x² + 8x has a common factor of 4x, so it factorises to 4x(x + 2).
Take out the whole common factor, letters as well as numbers. Here the question only needs a number: 4x + 8 = 4(x + 2).
3(x − 2) = 3x − 6.
The sign inside the bracket belongs to that term.
The term outside multiplies every term inside. 4(2x-3)=8x-12. Multiplying only the first term leaves the -3 untouched and gives 8x-3, which is a different expression. Draw arrows from the outside term to each inside term if you lose track, and write both products before you tidy. Count the arrows. A bracket with two terms inside needs two products. A bracket with three terms needs three. If you have written fewer products than terms, the expansion is not finished.
A minus outside the bracket changes every sign inside. -2(x-4)=-2x+8. The second product is -2×(-4)=+8, not -8. People remember to attach the minus to the first term and forget that it also hits the second. Read "minus two times minus four" as a positive. Then simplify if like terms appear from a second bracket. Write the positive product as +8 rather than leaving a double minus in the line, so the next collection does not flip it back.
Method: 1. Multiply the term outside the bracket by every term inside it. 2. If you multiply only the first term, 4(2x-3) is wrongly left as 8x-3. 3. Write both products before you tidy them.
(x+5)(x-2) has four products: x× x, x×(-2), 5× x and 5×(-2). That is x^2-2x+5x-10, which collects to x^2+3x-10. A grid with x and +5 along the side and x and -2 along the top keeps the four products visible. Two products, the "ends only", miss the middle term.
A squared bracket is a double bracket in disguise. (x+3)^2=(x+3)(x+3)=x^2+6x+9. It is not x^2+9. The middle term is 3x+3x. The same happens with a minus: (x-3)^2=x^2-6x+9. After any double expansion, "simplify" means collect the like terms. Leaving -2x+5x uncollected drops the final mark.
Method: 1. From (x+5)(x-2), write the four products x^2, -2x, 5x and -10. 2. Collect the like terms -2x+5x to reach x^2+3x-10. 3. A grid with x and +5 down the side, and x and -2 across the top, keeps those four products visible.
An expression such as 3(x+4)+2(x-5) is two single brackets added together. Expand each one on its own line before you collect. The first is 3x+12. The second is 2x-10. Together that is 3x+12+2x-10, which simplifies to 5x+2. If the second bracket is subtracted, the minus sits in front of the whole second expansion. 3(x+4)-2(x-5) becomes 3x+12-2x+10, because -2×(-5)=+10, and that collects to x+22.
The same pattern appears in a perimeter with an algebraic side and a numerical side, or when a question says "expand and simplify" without showing a double bracket. Treat every bracket as its own multiplication, write the results in a row, then collect like terms once. Collecting too early, while a bracket is still closed, is how a minus gets stuck to only the first term.
Method: 1. Treat 3(x+4)+2(x-5) as two separate single brackets added together. 2. Expand each bracket on its own line, then collect like terms. 3. The first bracket expands to 3x+12.
Look at the numbers first, then the letters. 6x^2+9x has a common factor of 3 in the numbers and one x in the letters, so the highest common factor is 3x. Divide each term by 3x: 6x^2÷ 3x=2x and 9x÷ 3x=3. The factorised form is 3x(2x+3). Check by expanding: 6x^2+9x, which matches. If the check produces a different first term, the factor outside is too big or too small. Adjust it and divide again. The bracket should look simpler than the line you started with. Do that check every time.
"Factorise fully" fails if a common factor is still inside. 2(6x^2+9x) is a true factorisation, but 6x^2 and 9x still share 3x. Taking out 6x is too much: 9x÷ 6x is not a whole number, so 6x is not a factor of both terms. Divide each original term by the factor you chose. 3x(2x+3) expands back to 6x^2+9x. If the inside terms still share a factor, you have not finished.
Method: 1. Look for a shared number first, then a shared letter. 2. In 6x^2+9x the numbers share 3 and the letters share one x, so take out 3x. 3. Divide each term by 3x to leave 2x and 3 inside the bracket.
Many lost marks in Expand and simplify questions come from a minus sign in front of a bracket. A minus outside multiplies every term inside, and it changes the sign of each one. So -2(1 - 2p) becomes -2 + 4p, because -2 × -2p is positive. Write each product on its own, with its sign, before you add anything together.
A question such as 5(p + 3) - 2(1 - 2p) has two brackets joined by a minus. Expand the first bracket to get 5p + 15. Expand the second, keeping the minus attached to the 2, to get -2 + 4p. Only then collect like terms. Put a ring or an arrow on each sign if that helps you keep track under exam pressure.
Method: 1. Circle the number and the sign in front of each bracket. 2. Multiply every term inside by that signed number, one term at a time. 3. Write out the expanded terms in a single line. 4. Collect the p terms, then the numbers, and write the answer in simplest form.
Tip: Two minus signs multiplied give a plus. This is where most of the slips happen.
The word fully means you must take out the highest common factor, and that includes letters as well as numbers. In 9b - 3b^2 both terms contain a 3 and a b, so the common factor is 3b. The answer is 3b(3 - b). If you only take out 3 you get 3(3b - b^2) and lose a mark, because the bracket still has a common factor of b.
Find the factor by looking at the numbers first, then each letter in turn. For each letter take the lowest power that appears in every term. Divide each term by the common factor to fill the bracket. Always check by expanding your answer. If the expansion gives the original expression, your factorisation is right. If a term inside the bracket still shares a factor, go again.
Method: 1. Find the highest common factor of the numbers. 2. Find each letter that appears in every term, with its lowest power. 3. Write the common factor outside and divide each term to get the bracket. 4. Expand your answer to check it returns the original expression.
Tip: If the bracket you have written still has a number or a letter in every term, you have not factorised fully.
Brackets with a letter outside, such as 4x^2(3x + 5), use the index laws. Multiply the numbers, then add the powers of the same letter. So 4x^2 × 3x = 12x^3 because x^2 × x = x^3, and 4x^2 × 5 = 20x^2. The result is 12x^3 + 20x^2. Treat a letter with no power as having power one.
Real exams also ask you to expand products like x(x - 4) or 4e(e + 2) for one mark each, and these give x^2 - 4x and 4e^2 + 8e. A quick numerical check saves marks. Pick a small value such as x = 2, work out the original and the expanded form, and see whether they match. If they differ, you have an error to find.
Method: 1. Multiply the numbers in front of each pair of terms. 2. Add the powers when the same letter is multiplied. 3. Write the sign of each product before moving on. 4. Check with a small number such as x = 2 in the original and the answer.
In a Show that question the answer is given, and your job is to reach it with full algebra. Expand each bracket on its own line, subtract carefully, and finish with the exact expression stated. Do not work backwards from the answer. Marks are awarded for each correct expansion, so show them clearly.
Spot the error questions give you a worked solution with one mistake in it. Read the lines in order and find the first line that is wrong. Usually it is a sign that did not change, or a term that was multiplied but not all of them. Say what is wrong in a short sentence, for example "-3 × -1 should be +3", and then give the corrected answer.
Method: 1. Expand each bracket separately and write the result in full. 2. Subtract the second expansion by changing every sign inside it. 3. Collect like terms and compare with the stated result. 4. In an error question, find the first wrong line and say exactly what should have happened.
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