ALGEBRA 6 - FACTORISING
Factorising is the reverse process of expanding. You may have noticed that expanding removed the brackets of an expression. So factorising puts the brackets in. To factorise an expression we need to form the brackets with an operator symbol inside (+ or - ). So the first thing we do is write down a pair of brackets. This is sometimes called 'monkey business' a term used to describe setting up the framework. Next, find a common factor to use as the multiplier outside the bracket. Let's look at an example or two.
Example (a) Factorise 5a + 5b
1) Form the brackets ( + ) we know it must be a '+' sign as the operator inside the brackets is a '+'.
2) 5a and 5b have a common factor of 5 so write 5 in front of the brackets.
so 5a + 5b = 5(a + b)
Example (b) Factorise 5x + 20
1) Form brackets ( + ) ...some teachers call this "monkey business"
2) The common factor between 5x and 20 is 5 because 5 x 4 = 20
so 5x + 20 = 5(x + 4)
You can check your answer by expanding the result. You should get 5 x x + 5 x 4 = 5x + 20
Example (a) Factorise 5a + 5b
1) Form the brackets ( + ) we know it must be a '+' sign as the operator inside the brackets is a '+'.
2) 5a and 5b have a common factor of 5 so write 5 in front of the brackets.
so 5a + 5b = 5(a + b)
Example (b) Factorise 5x + 20
1) Form brackets ( + ) ...some teachers call this "monkey business"
2) The common factor between 5x and 20 is 5 because 5 x 4 = 20
so 5x + 20 = 5(x + 4)
You can check your answer by expanding the result. You should get 5 x x + 5 x 4 = 5x + 20
ACTIVITY 1
Your turn.
Factorise:
1) 4a + 4b =
2) 10r + 10t =
3) 3a + 3b + 3c =
4) 6m + 24 =
5) 3g - 3 =
6) 4p - 16 =
7) 2x + 100 =
8) 6x - 18 =
9) 9x - 27 =
10) 5k + 55 =
Answers: 1) 4(a+b), 2) 10(r+t), 3) 3(a+b+c), 4) 6(m+4), 5) 3(g-1), 6) 4(p-4),
7) 2(x+50), 8) 6(x-3), 9) 9(x-3), 10) 5(k+11)
Your turn.
Factorise:
1) 4a + 4b =
2) 10r + 10t =
3) 3a + 3b + 3c =
4) 6m + 24 =
5) 3g - 3 =
6) 4p - 16 =
7) 2x + 100 =
8) 6x - 18 =
9) 9x - 27 =
10) 5k + 55 =
Answers: 1) 4(a+b), 2) 10(r+t), 3) 3(a+b+c), 4) 6(m+4), 5) 3(g-1), 6) 4(p-4),
7) 2(x+50), 8) 6(x-3), 9) 9(x-3), 10) 5(k+11)
Activity 2:
Answers:
ACTIVITY 3
Harder types:
Harder types:
ANSWERS:
factorising quadratic equations
Previously we learned that if we expand (x + 2)(x + 3) we get:
Compare this with something like
5 × 2 = 10
Here, the two numbers are called factors of 10 since 5 × 2 = 10 and 10 is called the product. Technically the '5' is called the multiplier and the '2' the multiplicand. You don't have to remember that but you should have an awareness of this:
5 × 2 = 10
Here, the two numbers are called factors of 10 since 5 × 2 = 10 and 10 is called the product. Technically the '5' is called the multiplier and the '2' the multiplicand. You don't have to remember that but you should have an awareness of this:
Previously we expanded double brackets to get a quadratic expression. That is, an expression with an x squared term in it. From the last section we learned that the word 'quadratic' is derived from the Latin quadratus which means 'square'. Here is the example from the previous page. Remember, factorising is the reverse process of expanding.
To factorise the above expression, follow these steps.
1) Form brackets (monkey business).
Because this example uses positive numbers we can add the operator signs too.
(x + )(x + )
2) Look for factors of 6 that add to make 5, the coefficient in front of the x. Make a list if that helps and then choose the one that works.
6 and 1
3 and 2
but 6 + 1 = 7 so that's not it
however, 3 + 2 = 5 yes! so choose that set of factors
It works because 3 x 2 = 6 and 3 + 2 = 5.
3) Complete the process
(x + 3)(x + 2)
To check our answer we could expand our result to get back where we started.
1) Form brackets (monkey business).
Because this example uses positive numbers we can add the operator signs too.
(x + )(x + )
2) Look for factors of 6 that add to make 5, the coefficient in front of the x. Make a list if that helps and then choose the one that works.
6 and 1
3 and 2
but 6 + 1 = 7 so that's not it
however, 3 + 2 = 5 yes! so choose that set of factors
It works because 3 x 2 = 6 and 3 + 2 = 5.
3) Complete the process
(x + 3)(x + 2)
To check our answer we could expand our result to get back where we started.
Another example:
1) Monkey business: (x + )(x + )
2) Look for factors of the constant 12 that add to give 7, the coefficient in front of the x. If you can't spot the answer right away that's OK, make up a list of factors of 12, the answer will be in there somewhere.
12 and 1, but they add to give 13 (we want 7)
6 and 2, but they add to give 8
4 and 3, they add to give 7 so that's it!
so our answer is (x + 4)(x +3)
Once again you can expand this out using FOIL or farmers field and you will get back to the above expression. Let's look at a negative example.
2) Look for factors of the constant 12 that add to give 7, the coefficient in front of the x. If you can't spot the answer right away that's OK, make up a list of factors of 12, the answer will be in there somewhere.
12 and 1, but they add to give 13 (we want 7)
6 and 2, but they add to give 8
4 and 3, they add to give 7 so that's it!
so our answer is (x + 4)(x +3)
Once again you can expand this out using FOIL or farmers field and you will get back to the above expression. Let's look at a negative example.
Monkey business: (x + )(x - )
I have included the + and - signs because one of the two factors of -15 will be positive and the other, negative. You don't have to do this now, you can do it later.
Look for factors of -15 that add to give -2.
With negatives there are much more options to choose from, but only one will work.
-15 and 1, that adds to give -14, nah!
15 and -1, that adds to give 14, so no.
3 and -5, that adds to give -2, looks like it...and
-3 and 5, adds to give 2, ah no.
So write the answers in the bracket.
I have included the + and - signs because one of the two factors of -15 will be positive and the other, negative. You don't have to do this now, you can do it later.
Look for factors of -15 that add to give -2.
With negatives there are much more options to choose from, but only one will work.
-15 and 1, that adds to give -14, nah!
15 and -1, that adds to give 14, so no.
3 and -5, that adds to give -2, looks like it...and
-3 and 5, adds to give 2, ah no.
So write the answers in the bracket.
Answer: (x + 3)(x - 5) again you can expand this out to check your answer using FOIL or other methods.
Activity 4
Now your turn, factorise:
Activity 4
Now your turn, factorise:
Answers: 1) (x + 4)(x + 1), 2) (x+ 10)(x - 2), 3) (x + 7)(x - 3), 4) (x - 3)(x + 2), 5) (x + 3)(x - 2)
6) (x + 10)(x + 3), 7) (x - 6)(x + 3), 8) (x + 9)(x + 1)
6) (x + 10)(x + 3), 7) (x - 6)(x + 3), 8) (x + 9)(x + 1)
Activity 5: Factorise:
Answers:
PERFECT SQUARES
The table below lists a selection of perfect squares (square numbers) from 1 to 400
A perfect square is a number that can be expressed as the product (multiplication) of two equal integers. In other words, if you times two whole numbers together that are the same, like 4 x 4 or 12 x 12, you get a number that is called a square number or perfect square.
Quadratic Equations with Perfect Squares and
difference of two squares
If a quadratic equation factorises such that the factors are the same, the quadratic is also called a 'perfect square'. We will look at a couple of examples.
1) Monkey business (x )(x )
Factors of 9 that add to give -6
9 and 1 adds to give 10, no
3 and 3 adds to give 6, no
-9 and -1 adds to give -10, no but,
-3 and -3 adds to give -6, that's it.
Factors of 9 that add to give -6
9 and 1 adds to give 10, no
3 and 3 adds to give 6, no
-9 and -1 adds to give -10, no but,
-3 and -3 adds to give -6, that's it.
This means the factors are (x - 3)(x - 3). We could have arrived at this solution earlier by noticing that the constant at the end, the 9 is a square number. This would lead us to the second answer above with 3's as the factors of 9. They add to give positive 6 which is not what we are after. You would then try the negative factors, -3's which is the correct solution.
Look for factors of 16 that add to give 8. This time the factors that work are 4 and 4. Later we will learn that although there are two factors, because they are the same, we say they are repeated factors.
This would be written as:
This would be written as:
However, not every quadratic ending in a square number is a perfect square quadratic! For example:
Here, the factors are 8 and 2, and written in factorised form = (x + 2)(x + 8). So looking for the
square root of 16 to get 4 will not help here since 4 + 4 does not equal 10. You didn't believe it was going to be that easy did you?
ACTIVITY 5
Now your turn.
square root of 16 to get 4 will not help here since 4 + 4 does not equal 10. You didn't believe it was going to be that easy did you?
ACTIVITY 5
Now your turn.