# Another Way of Factorising Cubics.

Try a different method of factorising cubics.

In the previous page, we had introduced a method, we termed the quickest to factorise cubics. Now we give you another method, which is also convenient.

Remember, cubics have the form:

$${f(x)=ax^3+bx^2+cx+d}$$

Using the factor theorem, you will first need to find, $$f(x) = 0$$.Then multiply it by some quadratic expression ( $$\,mx^2 +nx+p)\,$$. The expression

Let us try a few examples:

1. Factorise $$f(x)=x^3-5x^2-8x+12$$

The steps:

• Find the factor, using factor theorem. This means try different values of $$x$$ that will make $$f (\,x)\, =0$$. In this case, $$x = 1$$ will make $$f (\,x)\, =0$$. The factor is therefore, $$(\,x-1)\,$$
• Now we must multiply this factor by the quadratic. $${f(\,x)\,=(\,x-1)\,(\,mx^2 +nx+p)\,}$$
• The coefficient of the $$x^2$$ of the quadratic equals the coefficient of $$x^3$$. Therefore, $$m=1$$.

• Find the sum of $$x$$ multiplied by $$nx$$ and $$-1$$ multiplied by $$mx^2$$ and equate them to $$bx^2$$ of the quadratic.

\begin{gather*} nx^2 + (-mx^2) = bx^2 \\ nx^2 -mx^2 = -5x^2 \\ nx^2 -x^2 = -5x^2 \\ nx^2 = -4x^2 \\ n = -4 \end{gather*}
• Multiply $$-1$$ by $$p$$ and equate it to $$d$$.

\begin{gather*} -p=d\\ -p=12\\ p=-12 \end{gather*}
• Now substitute the values in the expression

\begin{gather*} f(\,x)\,= x^3-5x^2-8x+12\\ f(\,x)\, = (\,x-1)\,(\,x^2-4x-12)\, \\ f(\,x)\, = (\,x-1)\,(\,x-6)\,(\,x+2)\, \end{gather*}
• The expression has been factororised.

All these can be done in one step. But it gets better only with more practice.

2. Factorise $$f(x)=-x^3+10x^2-17x-28$$

The factor is $$(\,x+1)\,$$.

By following the steps in (1), we get:

\begin{gather} m=-1\\ n=11\\ p=-28 \end{gather}

The expression becomes:

\begin{gather*} f(\,x)\,= -x^3+10x^2-17x-28\\ f(\,x)\, = (\,x+1)\,(\,-x^2+11x-28)\, \\ f(\,x)\, = (\,x+1)\,[\,-(\,x^2-11x+28)\,]\, \\ f(\,x)\, = (\,x+1)\,[\,-(\,x-7)\,(\,x-4)\,]\, \\ f(\,x)\, = -(\,x+1)\,(\,x-6)\,(\,x-4)\, \end{gather*}
3. If this method does not work for you, try another one here.