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 $x^3-17x^2+54x-8$ given that one of the factors is $\left(x-4\right)$ .

Solution

Given that $x-4$ is a factor we can write

$x^3-17x^2+54x-8=\left(x-4\right)\times \text{(a quadratic polynomial)}$

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

$x^3-17x^2+54x-8=\left(x-4\right)\left(a{x}^2+bx+c\right)$

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

$x^3-17x^2+54x-8=a{x}^3+\left(b-4a\right)x^2+\left(c-4b\right)x-4c$ 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 $-17=b-4a$   With $a=1$ we have $-17=b-4$ so $b$ must equal $-13$ . Finally, equating constant terms we find $-8=-4c$   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

$x^3-17x^2+54x-8=\left(x-4\right)\left(x^2-13x+2\right)$

Exercises

Factorise into a quadratic and linear product the given polynomial expressions

1. $x^3-6x^2+11x-6$ , given that $x-1$ is a factor
2. $x^3-7x-6$ , given that $x+2$ is a factor
3. $2x^3+7x^2+7x+2$ , given that $x+1$ is a factor
4. $3x^3+7x^2-22x-8$ , given that $x+4$ is a factor

Answer