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

k

so that

x^{2} + 2 x + k

is a factor of

2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6

. Also find all zeroes of the two polynomials.

Accepted Answer

Given:

$x^{2} + 2 x + k$ is a factor of $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ .

So, when $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ is divided by $x^{2} + 2 x + k$ , the remainder will be zero.

Let us perform the division as shown below:

$x^{2} + 2 x + k) 2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6 (2 x^{2} - 3 x - (8 + 2 k) 2 x^{4} + 4 x^{3} + 2 {kx}^{2} - - -__________________0 - 3 x^{3} - (14 + 2 k) x^{2} + 5 x + 6 - 3 x^{3} - 6 x^{2} - 3 kx + + +___________________________0 - (8 + 2 k) x^{2} + (5 + 3 k) x + 6 - (8 + 2 k) x^{2} - 2 (8 + 2 k) x - k (8 + 2 k) + + +__________________________________0 + (21 + 7 k) x + (2 k^{2} + 8 k + 6)$

Here, the remainder should be zero.

$\Rightarrow 21 + 7 k = 0$ ......(1) and $2 k^{2} + 8 k + 6 = 0$ .......(2)

Solving equation (1), we have $k = \frac{- 21}{7} = - 3$

From middle term factorization of equation (2), we have $2 (k + 3) (k + 1) = 0$

$\Rightarrow k = - 3, k = - 1$

So, $k = - 3$ is common,

Now, the first polynomial is $x^{2} + 2 x - 3$ .

From middle term factorization, we have $x^{2} + 2 x - 3 = (x + 3) (x - 1)$

$\Rightarrow x = - 3, 1$

The second polynomial is $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ .

Other factor of $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ is $2 x^{2} - 3 x - (8 + 2 k)$ . [From division]

Put $k = - 3$ , we have $2 x^{2} - 3 x - (8 + 2 k) = 2 x^{2} - 3 x - 2$

From middle term factorization, we have $2 x^{2} - 3 x - 2 = (2 x + 1) (x - 2)$

$\Rightarrow x = \frac{- 1}{2}, 2$

Hence, the required answers are:

$k = - 3$ ,

zeroes of polynomial $x^{2} + 2 x - 3 = - 3, 1$

zeroes of polynomial $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6 = - 3, 1, \frac{- 1}{2}, 2$

Find $k$ so that $x^{2} + 2 x + k$ is a factor of $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ . Also find all zeroes of the two polynomials.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find k so that x2+2x+k is a factor of 2x4+x3-14x2+5x+6. Also find all zeroes of the two polynomials.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find $k$ so that $x^{2} + 2 x + k$ is a factor of $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ . Also find all zeroes of the two polynomials.