Name: Division of a Trinomial by a Trinomial
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Can we divide a polynomial by another polynomial to get the quotient polynomial and the remainder polynomial? Let's explore this concept in this video.

Question

Find the other two zeroes of the polynomial

3 x^{4} - 23 x^{3} + 62 x^{2} - 68 x + 24

, if two of its zeroes are

2

and

3

.

Accepted Answer

Let $f (x) = 3 x^{4} - 23 x^{3} + 62 x^{2} - 68 x + 24$

As two zeroes of $f (x)$ are $2$ and $3$ , $(x - 2)$ and $(x - 3)$ are both factors of $f (x)$ .

$\Rightarrow (x - 2) (x - 3) = x^{2} - 5 x + 6$ is a factor of $f (x)$ .

Dividing $f (x)$ by $x^{2} - 5 x + 6$ , we get,

$\begin{matrix} 3 x^{2} - & 8 x + & 4 \\ x^{2} - 5 x + 6 & 3 x^{4} - & 23 x^{3} + & 62 x^{2} - & 68 x + & 24 \\ 3 x^{4} - & 15 x^{3} + & 18 x^{2} \\ 0 x^{4} - & 8 x^{3} + & 44 x^{2} - & 68 x & 24 \\ - & 8 x^{3} + & 40 x^{2} - & 48 x \\ + & 4 x^{2} - & 20 x + & 24 \\ + & 4 x^{2} - & 20 x + & 24 \\ 0 \end{matrix}$

$\Rightarrow f (x) = 3 x^{4} - 23 x^{3} + 62 x^{2} - 68 x + 24 = (3 x^{2} - 8 x + 4) (x - 2) (x - 3)$

To further factorize $f (x)$ , we factorize $(3 x^{2} - 8 x + 4)$ by splitting the middle term.

$3 x^{2} - 8 x + 4 = 3 x^{2} - 6 x - 2 x + 4$

$= 3 x (x - 2) - 2 (x - 2)$

$= (3 x - 2) (x - 2)$

Thus,

$f (x) = (3 x - 2) (x - 2) (x - 2) (x - 3)$

Thus, $(3 x - 2), (x - 2), (x - 2), (x - 3)$ are the factors of $f (x)$ .

Hence, the zeroes of $f (x)$ are $\frac{2}{3}, 2, 2, 3$

Find the other two zeroes of the polynomial $3 x^{4} - 23 x^{3} + 62 x^{2} - 68 x + 24$ , if two of its zeroes are $2$ and $3$ .

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find the other two zeroes of the polynomial 3x4-23x3+62x2-68x+24, if two of its zeroes are 2 and 3.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find the other two zeroes of the polynomial $3 x^{4} - 23 x^{3} + 62 x^{2} - 68 x + 24$ , if two of its zeroes are $2$ and $3$ .