How to Factorise Monic Quadratics Using the Cross Method
Key takeaways
- The cross method factorises x2 + bx + c by finding two numbers that add to b and multiply to a × c. When a = 1, that product is just c.
- The proper way is to split the middle term with those two numbers, then factorise by grouping. For x2 + 6x + 5 the numbers are 5 and 1, giving (x + 5)(x + 1).
- When a = 1, the shortcut puts the two numbers straight into the brackets, so x2 − 10x + 21 becomes (x − 7)(x − 3).
- Under the sign rules, a positive a × c means both numbers take the sign of b. A negative a × c means the signs differ and the bigger number takes the sign of b.
- Put the terms in order and take out any HCF before you draw the cross. 4x4 − 12x3 − 40x2 factorises to 4x2(x − 5)(x + 2).
In this video
- 0:00 Why trinomials need a new method
- 3:26 Standard form: a, b and c
- 6:08 Setting up the cross
- 8:02 Example 1: splitting the middle term
- 11:09 The shortcut when a = 1
- 12:20 Checklist and sign rules
- 14:21 Example 2: a negative middle term
- 17:30 Examples 3 and 4: a negative constant
- 20:31 Practice questions
- 24:01 Two variables and higher powers
- 29:21 Terms out of order and the HCF first
This lesson shows you how to factorise monic quadratics, which are trinomials in the form x2 + bx + c, using the cross method. It is for students who can already take out the HCF and factorise by grouping.
Why do you need a new method to factorise trinomials?
Watch this part from 0:22
A trinomial is a polynomial with three terms. If it has an HCF, always take that out first. Factorising by grouping needs four terms, so it won’t work on a trinomial straight away.
The cross method lets you split the middle term into two terms, which gives you four terms to group.
- Three terms and no HCF: use the cross method.
- Four terms: use grouping.
- If neither works, the expression can’t be factorised.
None of the four trinomials in the next section has an HCF. Only two terms contain the variable, and the squared term has a coefficient of 1, so the only common factor is 1.
What are a, b and c in a quadratic?
Watch this part from 3:26
The standard form of a quadratic is ax2 + bx + c. Write this down, because the cross method depends on it.
- a is the number in front of x2.
- b is the number in front of x.
- c is the constant, the number with no variable attached.
If no number is written in front of x2, a is 1. In this lesson a is always 1. The variable can be y or z instead of x.
Each coefficient carries its sign, so a minus in front of a term makes its coefficient negative. If the first term is −x2, take out −1. That leaves x2 and flips the other signs.
| Trinomial | a | b | c |
|---|---|---|---|
| x2 + 6x + 5 | 1 | 6 | 5 |
| x2 − 10x + 21 | 1 | −10 | 21 |
| y2 − y − 30 | 1 | −1 | −30 |
| z2 + 3z − 28 | 1 | 3 | −28 |
How does the cross method work?
Watch this part from 6:08
Every trinomial in this lesson has a = 1. There is a version of the method for when a isn’t 1, and it comes in the next lesson.
- Check the trinomial is in standard form and find a, b and c.
- Draw a cross with b at the top and a × c at the bottom.
- Find two numbers that add to b and multiply to a × c. Write them on the sides of the cross.
- Split the middle term, using those numbers as the coefficients.
- Factorise the four terms by grouping.
This is the proper way of doing it. Finding the two numbers gets easier with practice.
How do you split the middle term?
Watch this part from 8:02
Example 1: factorise x2 + 6x + 5.
The top of the cross is 6. The bottom is 1 × 5 = 5.
Start with the factors of the bottom number. 5 is prime, so its only factors are 1 and 5. 5 + 1 = 6 and 5 × 1 = 5, so the numbers are 5 and 1. Use them to split the 6x.
x2 + 6x + 5
= x2 + 5x + x + 5
= x(x + 5) + 1(x + 5)
= (x + 5)(x + 1)
Once the middle term is split, it is ordinary factorising by grouping, which the previous lesson in this series covers.
What is the shortcut for factorising when a = 1?
Watch this part from 11:09
When a = 1, the answer always has the form (x + m)(x + n), where m and n are the two numbers from the cross. Each number keeps its sign.
In Example 1 the numbers are +5 and +1, so you can go straight to (x + 5)(x + 1). If you get stuck with the shortcut, go back to the proper way.
What should you check before using the cross method?
Watch this part from 12:20
- Is it in order? The terms go from the highest power down to the constant.
- Is the first term positive? If it isn’t, take out a negative.
- Is there an HCF? Take it out first.
- How many terms are there? Four terms can be grouped. Three terms need the middle term split.
The sign rules
- If the bottom of the cross is positive, both numbers have the same sign as b.
- If the bottom is negative, the numbers have different signs and the bigger number takes the sign of b.
Keep these rules next to you while you work. You don’t need to memorise them. With practice you will start to see the patterns.
How do you factorise a quadratic with a negative middle term?
Watch this part from 14:21
Example 2: factorise x2 − 10x + 21.
It is in order, has no HCF and has three terms. The top of the cross is −10 and the bottom is 21, so both numbers are negative.
The factor pairs of 21 are 1 and 21, and 7 and 3. 7 + 3 = 10, so the numbers are −7 and −3. The shortcut gives (x − 7)(x − 3).
Splitting the middle term gives the same answer.
x2 − 10x + 21
= x2 − 7x − 3x + 21
= x(x − 7) − 3(x − 7)
= (x − 7)(x − 3)
If your teacher wants to see the grouping step in an assessment, which is common, show it and use the shortcut to check. In later topics like calculus, factorising is assumed knowledge and you will often do it in your head.
How do you factorise when the constant is negative?
Watch this part from 17:30
Example 3: factorise y2 − y − 30.
The top of the cross is −1 and the bottom is −30, so one number is negative and one is positive. Try the factor pair 6 and 5.
- −5 and 6 multiply to −30, but add to 1.
- −6 and 5 multiply to −30 and add to −1.
The numbers are −6 and 5, so the answer is (y + 5)(y − 6). Use the variable from the question.
You won’t always pick the right pair first. You may need to swap the signs or try a different pair, and that happens all the time.
Example 4: z2 + 3z − 28
The top is 3 and the bottom is −28. Try 7 and 4 from the factor pairs of 28. b is positive, so the bigger number is positive. 7 + (−4) = 3 and 7 × (−4) = −28.
The answer is (z + 7)(z − 4).
Can you factorise these trinomials on your own?
Watch this part from 20:31
The video sets five questions. Pause it, attempt each one, then check your answer before you move on.
| Question | Two numbers | Answer |
|---|---|---|
| x2 + 8x + 15 | 3 and 5 | (x + 3)(x + 5) |
| x2 + 5x − 24 | 8 and −3 | (x + 8)(x − 3) |
| x2 − 12x + 20 | −10 and −2 | (x − 10)(x − 2) |
| v2 − 4v − 32 | −8 and 4 | (v − 8)(v + 4) |
| x2 + 5x + 8 | none | does not factorise |
What if a trinomial doesn’t factorise?
For x2 + 5x + 8, the factor pairs of 8 add to 6 (4 and 2) or 9 (1 and 8). No pair adds to 5. Once you have checked every pair with both signs, write does not factorise.
How do you factorise a trinomial with two variables?
Watch this part from 24:01
Example: factorise x2 − 7xy + 12y2.
Focus on x, the variable with the highest power. The y is part of b, so b = −7y. Treat 12y2 as the constant.
Ignore the y for now. Two numbers that add to −7 and multiply to 12 are −3 and −4. Add the y back onto each one.
(x − 3y)(x − 4y)
Adding two y terms gives a y term, and multiplying two gives y2. That is why the y can be carried along.
How do you factorise a trinomial with higher powers?
Watch this part from 27:20
Example: factorise x4 + 5x2 + 6.
b and c work the same whatever the powers are. 3 + 2 = 5 and 3 × 2 = 6, so the numbers are 3 and 2.
x4 + 5x2 + 6
= x4 + 3x2 + 2x2 + 6
= x2(x2 + 3) + 2(x2 + 3)
= (x2 + 3)(x2 + 2)
The shortcut still works, but you have to think about how the powers relate. It is often easier to split the middle term and group.
What if the terms are out of order?
Watch this part from 29:21
Example: factorise 18 − 19w + w2.
Rewrite it in order first, keeping each term’s sign, so b and c land in the right places on the cross.
w2 − 19w + 18
The top is −19 and the bottom is 18. 1 + 18 = 19, and both numbers must be negative, so they are −18 and −1.
(w − 18)(w − 1)
Why should you take out the HCF first?
Watch this part from 30:37
Example: factorise 4x4 − 12x3 − 40x2.
4 goes into 12 and 40. Every term has a power of x, so take out the lowest power, x2. The HCF is 4x2.
4x4 − 12x3 − 40x2
= 4x2(x2 − 3x − 10)
Inside the brackets a = 1 and there is now a constant. −5 + 2 = −3 and −5 × 2 = −10, so the numbers are −5 and 2.
= 4x2(x − 5)(x + 2)
Keep the HCF outside the brackets, usually at the front. Whatever method you use, take out the HCF first if you can. The trinomial left inside usually factorises again.
The next lesson covers trinomials where a isn’t 1 and no number can be taken out.
Frequently asked questions
What is the cross method for factorising?
It is a way of organising your thinking when you factorise a trinomial. You draw a cross with b at the top and a × c at the bottom, then find two numbers that add to b and multiply to a × c. Those numbers let you split the middle term and factorise by grouping.
Do I have to show the grouping step or can I just use the shortcut?
Show it if your teacher wants to see it, which is common in assessments. You can then use the shortcut to check your answer.
What do you write if a trinomial won't factorise?
Write does not factorise. Check every factor pair of the bottom number, with both signs, before you decide.
Still stuck on factorising?
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