2 Factorising polynomials and equating coefficients

We will consider how we might find the solution to some simple polynomial equations. An important part of this process is being able to express a complicated polynomial into a product of simpler polynomials. This involves factorisation .

Factorisation of polynomial expressions can be achieved more easily if one or more of the factors is already known. This requires a knowledge of the technique of ‘equating coefficients’ which is illustrated in the following example.

Example 23

Factorise the expression x 3 17 x 2 + 54 x 8 given that one of the factors is ( x 4 ) .

Solution

Given that x 4 is a factor we can write

x 3 17 x 2 + 54 x 8 = ( x 4 ) × (a quadratic polynomial)

The polynomial must be quadratic because the expression on the left is cubic and x 4 is linear. Suppose we write this quadratic as a x 2 + b x + c where a , b and c are unknown numbers which we need to find. Then

x 3 17 x 2 + 54 x 8 = ( x 4 ) ( a x 2 + b x + c )

Removing the brackets on the right and collecting like terms together we have

x 3 17 x 2 + 54 x 8 = a x 3 + ( b 4 a ) x 2 + ( c 4 b ) x 4 c Like terms are those which involve the same power of the variable ( x ).

Equating coefficients means that we compare the coefficients of each term on the left with the corresponding term on the right. Thus if we look at the x 3 terms on each side we see that x 3 = a x 3   which implies a must equal 1. Similarly by equating coefficients of x 2 we find 17 = b 4 a   With a = 1 we have 17 = b 4 so b must equal 13 . Finally, equating constant terms we find 8 = 4 c   so that c = 2 .

As a check we look at the coefficient of x to ensure it is the same on both sides. Now that we know a = 1 , b = 13 , c = 2 we can write the polynomial expression as

x 3 17 x 2 + 54 x 8 = ( x 4 ) ( x 2 13 x + 2 )

Exercises

Factorise into a quadratic and linear product the given polynomial expressions

  1. x 3 6 x 2 + 11 x 6 , given that x 1 is a factor
  2. x 3 7 x 6 , given that x + 2 is a factor
  3. 2 x 3 + 7 x 2 + 7 x + 2 , given that x + 1 is a factor
  4. 3 x 3 + 7 x 2 22 x 8 , given that x + 4 is a factor
  1. ( x 1 ) ( x 2 5 x + 6 ) ,
  2. ( x + 2 ) ( x 2 2 x 3 ) ,
  3. ( x + 1 ) ( 2 x 2 + 5 x + 2 ) ,
  4. ( x + 4 ) ( 3 x 2 5 x 2 ) .