Factorisation of trinomials of the form x2+bx+c
Trinomials are expressions with three terms.
For example, x2 + 14x + 49 is a trinomial. There is no single method by which all trinomials can be factorised. We need to study the pattern in trinomials and choose the appropriate method to factorise the given trinomial.
Factorising a trinomial by splitting the middle term
The general form of the trinomial is (x2 + cx + d) where c and d have different numerical values: c = a + b, and d = ab.
In these examples, study the relation between the middle and the last terms.
Let us study some examples.
Example 1:
Factorise a2 + 16a + 28.
Suggested answer:
Step 1: The factor pairs of the third term are ((7,4); (14,2); (28, 1).
Step 2: The pair of factors whose sum is equal to the co-efficient ofthe middle term is 14 and 2.
Step 3: Rewrite the expression using these factors
Remember:
In the given trinomial expression if all terms are positive, then both the factors are positive.
Example 2:
Factorise x2 - 15x + 14.
Suggested answer:
Step 1: The factor pairs of the last term are (7,2), (14,1), (-1,-14).
Step 2: The pair of factors whose sum is equal to -15, the co- efficient of the middle term is (-1,-14).Step 3: Rewrite the expression using these factors
Step 4: Group the terms and factorise.
Remember:
If the middle term is negative, and last term is positive, signs of both the factors will be negative.
Example 3:
Factorise w2 + 4w - 21.
Suggested answer:
Step 1: Factor pairs of the last term are (-7,+3), (21,-1), (+7,-3),
(-21,+1).Step 2: The pair of factors whose sum is equal to the co-efficient of
the middle term is (+7,-3).Step3: Rewrite the expression using these factors as
w2 + 7w - 3w -21.Step 4: Group the terms and factorise.
Remember:
If the middle term is positive, and the last term is negative, the sign of one of the factors is positive and the other is negative.
Example 4:
Factorise y2 - 3y - 18.
Suggested answer:
Step 1: Factor pairs of the last term are
(9,-2); (-6,3); (-18,1);(-9,2);(6,-3); (18,-1).Step 2: The pair of factors whose sum is equal to the co-efficient of
the middle term is (-6,3).Step 3: Rewrite the expression using these factors as
Step 4: Group the terms and factorise.
Steps to factorise a trinomial of the form x2 + bx + c where b and c are integers:
- Find all pairs of factors whose product is the last term of the trinomial.
- From the pairs of factors from step 1, choose a pair of factors whose sum is the coefficient of the middle term of the trinomial.
- Split the middle term using the pair of factors from step 2 and rewrite the trinomial.
- Group the terms from step 3 and factorise.
- Verify the solution.
To factorise expressions of the type x2 + bx + c, you will find two numbers a and b such that their sum is equal to the coefficient of the middle term and their product is equal to the last term(constant).
A similar procedure is used to factorise expressions of the 
Example 1:
Factorise 6x2 + 7x + 2.
Suggested answer:
(1,12), (3,4), (2,6) are the factor pairs of 12.
The factor pair (3,4) is such that 3 + 4 = 7and 3 x 4 = 12 i.e, sum of the numbers is equal to the coefficient of the middle term(b), and their product is equal to the product of the coefficient of the first and the last term (a.c).Split the middle term and rewrite the expression,
Group the terms and factorise = 3x(2x+1) + 2(2x+1).
Verify the result by actual multiplication.
Verification by actual multiplication:
Example 2:
Factorise 5x2 - 9x - 18.
Suggested answer:
Verify by actual multiplication.
Verification by actual multiplication:
Example 3:
Factorise 3y2 - 16y - 12.
Suggested answer:
- Find the product (ac), of the coefficient of x2 and the last term.
- List the factor pairs of ac.
- Choose a factor pair whose sum is the coefficient of the middle term.
- Rewrite the polynomial by splitting the middle term.
- Regroup and factorise.
