Name: Rational Expressions: Operations and Manipulations
Uploaded: 2019-07-19
Duration: 12 min 15 s
Description: In this video, we will learn the concept of multiplying, simplifying and cancelling the common factors with detailed examples. View the video now.

Question

Find the GCD of the polynomials $x^{3} + x^{2} - x + 2$ and $2 x^{3} - 5 x^{2} + 5 x - 3$ .

Accepted Answer

The procedure for finding Greatest Common Divisor of two given polynomials $f (x)$ and $g (x)$ is

(i) First, divide $f (x)$ by $g (x)$ to obtain $f (x) = g (x) q (x) + r (x)$ where $q (x)$ is the quotient and $r (x)$ is the remainder. Then, $deg [r (x)] < deg [g (x)]$

(ii) If the remainder $r (x)$ is non-zero, divide $g (x)$ by $r (x)$ to obtain $(x) = r (x) q (x) + r_{1} (x$ where $r_{1} (x)$ is the new remainder. Then, $deg [r_{1} (x)] < deg [r (x)]$ . If the remainder $r_{1} (x)$ is zero, then $r (x)$ is the required GCD.

(iii) If $r_{1} (x)$ is non-zero, then continue the process until we get zero as the remainder. The divisor at this stage will be the required GCD. We write $G C D [f (x), g (x)]$ to denote the GCD of the polynomials $f (x), g (x)$ .

Let $f (x) = 2 x^{3} - 5 x^{2} + 5 x - 3$ and $g (x) = x^{3} + x^{2} - x + 2$

$\begin{matrix} x^{3} + x^{2} - x + 2 & 2 x^{3} & - & 5 x^{2} & + & 5 x & - & 3 \\ 2 x^{3} & + & 2 x^{2} & - & 2 x & + & 4 \\ - & - & + & - \\ - & 7 x^{2} & + & 7 x & - & 7 \end{matrix}$

$= - 7 (x^{2} - x + 1)$

$- 7 (x^{2} - x + 1) \neq 0$ , note that $- 7$ is not a divisor of $g (x)$ .

Now, dividing $g (x) = x^{3} + x^{2} - x + 2$ by the new remainder $x^{2} - x + 1$ (leaving the constant factor), we get,

$\begin{matrix} x & + & 2 \\ x^{2} - x + 1 & x^{3} & + & x^{2} & - & x & + & 2 \\ x^{3} & - & x^{2} & + & x \\ - & + & - \\ 2 x^{2} & - & 2 x & + & 2 \\ 2 x^{2} & - & 2 x & + & 2 \\ - & + & - \\ 0 \end{matrix}$

Here, we get zero remainder

Therefore, $G C D (2 x^{3} - 5 x^{2} + 5 x - 3, x^{3} + x^{2} - x + 2) = x^{2} - x + 1$ .

Find the GCD of the polynomials $x^{3} + x^{2} - x + 2$ and $2 x^{3} - 5 x^{2} + 5 x - 3$ .

Important Questions on Algebra

Learn and Prepare for any exam you want !

Find the GCD of the polynomials x3+x2-x+2 and 2x3-5x2+5x-3.

Important Questions on Algebra

Learn and Prepare for any exam you want !

Find the GCD of the polynomials $x^{3} + x^{2} - x + 2$ and $2 x^{3} - 5 x^{2} + 5 x - 3$ .