Question

Find the factors of $f\left( x \right) = 2{x^4} + 7{x^3} - 4{x^2} - 27x - 18$ using the method of synthetic division.

Answer 1

Hint:
We are asked to find the factors of the function $f\left( x \right) = 2{x^4} + 7{x^3} - 4{x^2} - 27x - 18$.
Try putting the values of x as $ \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$ and check whether we get the value of function as 0 as 2 has factors 1 and 2, and 18 has factors 1, 2, 3, 6, 9 and 18.
Using hit and trial method, find for what values of x as mentioned above the given function becomes 0.
Thus, apply synthetic division to find the required factors.

Complete step by step solution:
We are asked to find the factors of the function $f\left( x \right) = 2{x^4} + 7{x^3} - 4{x^2} - 27x - 18$ .
In this question, we need to use a hit and trial method. So, let us put some values of x as some numbers and see whether we get the remainder as 0 at the particular value of x. If we get the value of function 0, then the number will be the root of the function.
We will first try the factors of the coefficients of the first and last term i.e. 2 and 18 respectively.
Since, 2 has factors 1 and 2, and 18 has factors 1, 2, 3, 6, 9 and 18.
So, we can try putting the values of x as $ \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$ and check whether we get the value of the function as 0.
By putting several values, we come to a conclusion that, by putting the values of x as -1, 2 and 3, we get the value of function as 0.
Thus, we will use the synthetic method of division to solve it further.
Now, for synthetic division, we will write the coefficients of variables in sequence and do the synthetic method using $x = - 1$ .
 

Thus, $x = - 1$ is a factor of the given function.
We will now do synthetic division further by taking $x = - 3$
 

Thus, $x = - 3$ is another factor of the given function.
Again, we will do synthetic division further by taking $x = 2$
 

Thus, we get $x = 2$ as another factor of the given function.
Also, we get the equation $2x - 3 = 0$, from the above synthetic division.

Thus, the factors of the given function $f\left( x \right) = 2{x^4} + 7{x^3} - 4{x^2} - 27x - 18$ are $\left( {x + 1} \right),\left( {x - 2} \right),\left( {x + 3} \right)$ and $\left( {2x - 3} \right)$.

Note:
Step-by-step method to do division using synthetic division:
Example: Divide ${x^2} + 5x + 6$ by \[X - 1\] .
To do so, first write the coefficients of the given polynomial as follows:

Now, let \[X - 1 = 0\] . Thus, \[X = 1\]. Put \[X = 1\] at left as:
 

Take the first number down and carry it down as it is below the division as:

Now, multiply the below division value by 1 on the left side and put the result in the next column as:

Adding the second column:

Now, multiplying below division 6 by left side 1 and adding it in the third column as:

Adding the third column:

The first digit in the below division part is the coefficient of x, the middle digit in the below division part is the constant term and the last digit in the below division part is the remainder.
Thus, on dividing ${x^2} + 5x + 6$ by \[X - 1\], we get the factor as \[X + 6\] and the remainder is 0.