Question

$\text{If}(x-3)\text{and}(x-2)\text{are two factors of}\text{the polynomial}{x}^3-6x^2+11x-6,\text{then find the third factor.}$

• A. $x-1$
• B. $x-4$
• C. $x-5$
• D. $x-6$

Answer 1

The correct option is A $x-1$

$\text{Since (x-3) and (x-2) are the factors}\text{of the given polynomial, the product}\text{of (x-3) and (x-2)}(x-3)(x-2)=x^2-2x-3x+6=x^2-5x+6\text{will also be a factor of the polynomial.}$

$\text{Now, for finding the third factor,}\text{divide the polynomial}{x}^3-6x^2+11x-6\text{by}(x^2-5x+6).$

$x^2-5x+6\stackrel{x-1}{)\overline{x^3-6x^2+11x-6}}\underset{–}{{}_{-}{x}^3\mp 5x^2\pm 6x+0}\underset{–}{-x^2+5x-6\mp x^2\pm 5x\mp 6}0$

$\therefore (x-1)\text{is the third factor.}$