You should be able to
factorise by grouping .
Always look through all examples in textbook before lesson.
RECAP: Factorisation by taking out common factors:
Single term as common factor:
ALL TERMS must have a common factor.
Look at it part by part.
Eg: Factorise 4ab - 2abc + 8ac
STEPS:
Look at the coefficient. Does it have a common factor for all terms. (For this example coefficients have a common factor 2.)
Next look at the variables individually. (ie asking yourself: Does a appear in all terms? Does b appear in all terms? Does c appear in all terms? For this example, a appears in all terms but b and c do not, so a is a common factor)
Therefore common factor for this is 2a.
Possible ways to find what is left in the bracket after you take out common factor
4ab - 2abc + 8ac = 2a( ? )
Method 1: every original term divide by the factor.
Method 2: Ask yourself: What must I multiply to 2a to get 4ab?
What must I multiply to 2a to get -2abc?
What must I multiply to 2a to 8ac?
NOTE: The number of terms left in the bracket will be the same as the original number of terms.
Therefore to factorise 4ab - 2abc + 8ac = 2a(2b - bc + 4c)
Brackets as common factor:
Common factors need not be a single term. It can be a bracket as well.
Eg: Facorise x(a + b) + y(a + b)
Important: To consider brackets as a common factor they must satisfy these conditions:
same terms in the brackets
same sign for all terms
(a + b) has common terms in both brackets.
(a + b) have the same sign for all terms in both brackets.
Hence (a + b) is a common factor.
Therefore to factorise x(a + b) + y(a +b) = (a + b)(x + y)
Example where brackets are not considered a common factor:
Eg: a(p + q) - b(p-q)
(p + q) and (p - q) has common terms in both brackets.
for (p + q) and (p - q), p have the same sign for both brackets BUT q does not. One q is positive while the other q is negative.
Hence (p + q) and (p - q) is a NOT a common factor.
Special Case Factorisation:
When there are the same terms in the brackets, but each term have exactly the opposite signs.
This type of question you will need to take out a common factor -1.
Simple example to take out common factor -1
Eg 1: Eg 2: Eg 3: Eg 4:
(a + b) (- a - b) (a - b) (-a + b)
= - (-a - b) = - ( a + b) = - (- a + b) = - (a - b)
NOTE: If you expand back, you should get back the original question
When you do not see a number in front of the bracket this means it is the number 1.
Eg:
r(e + f) - w(-e-f) Row 1
= r(e + f) + w(e + f) Row 2
= (e + f)(r + w) Row 3
Row 1: If you look at this question, both brackets have same term but exactly different sign for each of the term.
Row 2: Take out the negative 1 from the second bracket, sign will change in front of the second bracket, and the sign for the terms inside the second bracket will also change
Row 3: All terms in the bracket will have exactly the same sign and terms hence it has become a common factor. Then we take out common factor (e + f) and the factors left second bracket will be (r + w)
Factorisation by grouping:
There should be 4 terms.
Pair the terms. (ie: Try to pair them such that both pairs will have terms that have common factors)
There may be special cases where only one pair have common factors.
Eg: Factorsie ax + bx - ay - by
STEPS:
Pair the terms into 2 brackets linked by + . Note that all term's sign must follow into the bracket.
ie: (ax + bx) + (- ay - by)
Take out the common factors from both the brackets. (DO NOT do anything to the sign)
ie: (ax + bx) + (- ay- by)
= x(a + b) + y(- a - b)
Note if the terms in the brackets are special terms. (ie: they have the same terms but exactly opposite signs) If they are special terms, you will need to take out -1 and all terms in the bracket will change sign.
ie: (ax + bx) + (- ay- by)
= x(a + b) + y(- a - b)
=x(a + b) - y (a + b)
Check that the terms and signs of the terms in the brackets are the same. Then take out the bracket as the common factor.
ie: (ax + bx) + (- ay- by)
= x(a + b) + y(- a - b)
=x(a + b) - y (a + b)
=(a + b)(x - y)