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 polynomial of least degree which should be subtracted from the polynomial

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

so that it is exactly divisible by

x^{2} - x + 1

.

Accepted Answer

According to the division algorithm of polynomial, if $f (x)$ and $g (x)$ are two polynomials where $g (x) \neq 0$ , then we can write $f (x) = g (x) \times q (x) + r (x)$ , where $q (x)$ is the quotient and $r (x)$ is the remainder after division.

Clearly, if $r (x)$ [having the least degree compared to the dividend, the divisor and the quotient] is subtracted from $f (x)$ , then $f (x)$ is exactly divisible by $g (x)$ .

Here, $f (x) = x^{4} + 2 x^{3} - 4 x^{2} + 6 x - 3$ and $g (x) = x^{2} - x + 1$ .

We shall divide $x^{4} + 2 x^{3} - 4 x^{2} + 6 x - 3$ by $g (x)$ and find $r (x)$ as follows:

$\begin{matrix} x^{2} & + & 3 x & - & 2 \\ x^{2} - x + 1 & x^{4} & + & 2 x^{3} & - & 4 x^{2} & + & 6 x & - & 3 \\ x^{4} & - & x^{3} & + & x^{2} \\ - & + & - \\ + & 3 x^{3} & - & 5 x^{2} & + & 6 x & - & 3 \\ + & 3 x^{3} & - & 3 x^{2} & + & 3 x \\ - & + & - \\ - & 2 x^{2} & + & 3 x & - & 3 \\ - & 2 x^{2} & + & 2 x & - & 2 \\ + & - & + \\ x & - & 1 \end{matrix}$

Quotient $q (x) = x^{2} + 3 x - 2$

Remainder $r (x) = x - 1$ .

Therefore, the $x - 1$ must be subtracted from the polynomial $f (x) = x^{4} + 2 x^{3} - 4 x^{2} + 6 x - 3$ so that the resulting polynomial is exactly divisible by $x^{2} - x + 1$ .

Find the polynomial of least degree which should be subtracted from the polynomial $x^{4} + 2 x^{3} - 4 x^{2} + 6 x - 3$ so that it is exactly divisible by $x^{2} - x + 1$ .

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find the polynomial of least degree which should be subtracted from the polynomial x4+2x3-4x2+6x-3 so that it is exactly divisible by x2-x+1.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Find the polynomial of least degree which should be subtracted from the polynomial $x^{4} + 2 x^{3} - 4 x^{2} + 6 x - 3$ so that it is exactly divisible by $x^{2} - x + 1$ .