Name: Application of the Remainder Theorem: Example 2
Uploaded: 2019-07-19
Duration: 4 min 59 s
Description: What are the conditions to be a factor of a polynomial? Can you guess what the condition for x+1 to be a Factor of a Polynomial is? Watch the video to learn...

Question

Use Remainder Theorem to factorise the following polynomial completely:

3 x^{3} + 2 x^{2} - 19 x + 6

Accepted Answer

The remainder theorem states that if a polynomial $f (x)$ is divided by $(x - α)$ , then the remainder is $f (α)$ .

If the remainder $f (a) = 0$ , then $(x - a)$ is a factor of the polynomial $f (x)$ .

Here, let $f (x) = 3 x^{3} + 2 x^{2} - 19 x + 6$ .

Putting $x = 2$ (We obtain this value of $x$ by using the trial and error method) in $f (x)$ , we obtain, $f (2) = 3 \times {(2)}^{3} + 2 \times {(2)}^{2} - 19 \times 2 + 6$

$= 3 \times 8 + 2 \times 4 - 38 + 6$

$= 24 + 8 - 38 + 6$

$= 38 - 38$

$= 0$ .

Hence, $(x - 2)$ is one factor of $f (x) = 3 x^{3} + 2 x^{2} - 19 x + 6$ .

For the other factors, we shall divide $f (x) = 3 x^{3} + 2 x^{2} - 19 x + 6$ by $x - 2$ using the long division method.

$\begin{matrix} 3 x^{2} & + & 8 x & - & 3 \\ x - 2 & 3 x^{3} & + & 2 x^{2} & - & 19 x & + & 6 \\ 3 x^{3} & - & 6 x^{2} \\ - & + \\ + & 8 x^{2} & - & 19 x & + & 6 \\ + & 8 x^{2} & - & 16 x \\ - & + \\ - & 3 x & + & 6 \\ - & 3 x & + & 6 \\ + & - \\ 0 \end{matrix}$

Quotient $= 3 x^{2} + 8 x - 3$

Remainder $= 0$ .

So, the other factor of $3 x^{3} + 2 x^{2} - 19 x + 6$ is $3 x^{2} + 8 x - 3$ .

Now, we have, $3 x^{2} + 8 x - 3 = 3 x^{2} + 9 x - x - 3$

$= 3 x (x + 3) - 1 (x + 3)$

$= (x + 3) (3 x - 1)$ .

Therefore, we have, $3 x^{3} + 2 x^{2} - 19 x + 6 = (x + 3) (x - 2) (3 x - 1)$ .

Use Remainder Theorem to factorise the following polynomial completely:
$3 x^{3} + 2 x^{2} - 19 x + 6$

Important Questions on Factorisation

Learn and Prepare for any exam you want !

Use Remainder Theorem to factorise the following polynomial completely: 3x3+2x2-19x+6

Important Questions on Factorisation

Learn and Prepare for any exam you want !

Use Remainder Theorem to factorise the following polynomial completely:
$3 x^{3} + 2 x^{2} - 19 x + 6$