Factorising trinomials has long been one of the thorns in students' sides. If the concept is not grasped in grade 9 (when it is taught in South Africa), it can make the rest of a maths high school career difficult. In this article I will show you 5 steps to easily factorise trinomials when a = 1.
The standard form of a trinomial
ax^2 + bx +c Remember that you need to write your equation in standard form before you can factorise it. This means that your highest exponent comes first, and the exponents are written in descending order. Once we have written our expression or equation in standard form we can move on to the next step.Looking at the signs
With trinomials, we need to pay attention to the signs. They tell us what our factors (brackets) will look like. When the sign in front of c is + (positive) we know that the signs in both brackets will be the same. This is because + x + or - x - is a +. We look at the sign in front of b to tell us whether to use + in both brackets (when b is positive), or - in both brackets (when b is negative). When the sign in front of c is - (negative) it means that the signs in our brackets will be different. This is because + x - or - x + gives us a -. The sign in front of c also tells us what to do with our factors of c. When c is positive, we add our factors together, and when c is negative we subtract the one factor from the other one.5 Steps to Easily Factorise Trinomials
- Step 1: Put your equation into standard form. This means checking for the highest common factor too.
- E.g. 10x + 21 + x^2 becomes x^2 +10x +21
- Step 2: Find all the factors for the value of c
- E.g. Factors of 21 are 1 and 21; and 3 and 7, 7 and 3, and 21 and 1.
- Step 3: Look at the sign in front of c to decide what signs to use.
- E.g. In our example c = 21 is positive, and the sign in front of b is positive so our brackets will both have a +.
- We can write them down as (x + ◊)(x + ◊)
- Step 4: We look at the sign in front of c to decide what to do with our factors. When c is positive, we are looking for factors that add together to give us the value of b. When c is negative, we subtract the one factor from the other to find the difference that is equal to b.
- E.g. 1 + 21 = 22 which is not 10 so these are not our factors.
- 3 + 7 = 10 which is b, so these two numbers are our factors.
- We can write our brackets as (x+3)(x+7)
- Step 5: Quick check. It is always a good idea to make sure that your two factors now written in the brackets multiply to give c, and add (whether by sum or difference) to give b.
- 3 x 7 = 21 √
- 3 + 7 = 10 √
- We are happy with our sum and can move on 🙂
Some more Examples:
Example 1: Factorise:
30 + 11x +x^2- Step 1: Make sure the trinomial is in standard form:
- x^2 + 11x + 30
- There is no common factor to take out.
- Step 2: Find the factors of c:
- 1 and 30 and 30 and 1
- 2 and 15 and 15 and 2
- 3 and 10 and 10 and 3
- 5 and 6 and 6 and 5
- Step 3: Choosing our signs
- In the expression, 30 is positive, so that means that our brackets are going to have the same signs.
- In the expression, 11x is positive, so that means that both our brackets will have + signs.
- We can write our brackets as (x + ◊)(x +◊)
- Step 4: Add our factors to find b
- 1 + 30 = 31 and 30 + 1 = 31 not b = 11
- 2 + 15 = 17 and 15 + 2 = 17 not b = 11
- 3 + 10 = 13 and 10 +3 = 13 not b = 11
- 5 + 6 = 11 and 6 + 5 = 11 yes, b = 11 - these are our factors
- So we can write: (x+5)(x+6)
- Step 5: Quick check:
- 5 + 6 = 11 √
- 5 x 6 = 30 √
Example 2: Factorise:
x^2 - 9x + 14- Step 1: Check for standard form and the highest common factor
- Our expression is already in standard form and there is no highest common factor so we can move on to step 2.
- Step 2: Find the factors of c:
- 1 and 14 and 14 and 1
- 2 and 7 and 7 and 2
- Step 3: Choosing our signs:
- The sign in front of c is +, so our brackets will have the same signs.
- The sign in front of b is -, so our brackets will both have - signs.
- We can write our brackets as (x - ◊)(x - ◊)
- Step 4: Add our factors to find b
- -1 + -14 = -15 and -14 + -1 = -15
- -2 + -7 = -9 and -7 + -2 = -9 - this is our value of b so we have found our factors.
- We can now write: (x - 2)(x - 7)
- Step 5: Quick check:
- -2 + -7 = -9 √
- -2 x -7 = 14 √
Example 3: Factorise:
x^2 -3x - 18- Step 1: Check for standard form and highest common factor
- The expression has no highest common factor (other than 1), and is already in standard form so we can move on to step 2.
- Step 2: Find the factors of c:
- 1 and 18 and 18 and 1
- 2 and 9 and 9 and 2
- 3 and 6 and 6 and 3
- Step 3: Choosing our signs:
- The sign in front of c is negative (-), so that means our brackets will have different signs.
- When this happens, we dont write the signs down yet, we wait to see which sign will belong to which factor.
- Step 4: Add our factors to get b:
- Now because our signs are different, one factor in each pair will get a negative sign, and the other will get a positive sign.
- +1 - 18 = -17
- +2 - 9 = -7
- +3 - 6 = -3 this is our b value, so we don't need to continue adding (or subtracting) our factors.
- We can now write our brackets like this: (x + 3)(x - 6)
- Step 5: Quick check:
- +3 - 6 = -3 √
- +3 x -6 = -18 √
Example 4: Factorise:
2x^2 +4x - 160 In this example I am simply going to list each step without a further explanation. See if you can understand why each step happened. If you can, you are on your way and ready to start practicing on your own 🙂- Step 1: 2(x^2 + 2x - 80)
- Step 2: 1 and 80
- 2 and 40
- 4 and 20
- 5 and 18
- 8 and 10
- Step 3: c is negative, so our brackets will have different signs.
- Step 4: -1 + 80 = 79
- -2 + 40 = 38
- -4 + 20 = 16
- -5 + 18 = 13
- -8 + 10 = 2 √
- We can write: 2(x - 8)(x + 10)
- Don't forget to include your common factor in the last step. 😉
- Step 5: Quick check
- -8 + 10 = 2 √
- -8 x 10 = -80 √
Example 5: Factorise:
3x^3 - 36x^2 + 81x For this example I am simply going to give the solution. Try the answer on your own before you take a peak at my answer. 🙂 3(x - 3)(x - 9)Now that you've conquered the 5 steps to easily factorise trinomials with a = 1, you can practice these worksheets:
- Worksheet on Factorising Trinomials when a = 1 and b and c are positive
- Factorising Trinomials when a = 1, b is negative and c is positive
- Worksheet with Memo: Factorising Trinomials when a = 1 and c is negative