Tag: multiplication and division of algebraic expressions

Questions Related to multiplication and division of algebraic expressions

The sum of the digits of a 3 digit number is subtracted from the number. The resulting number is always.

maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

  1. Divisible by 6

  2. Not divisible by 6

  3. Divisible by 9

  4. Not divisible by 9

Show answer
Correct Option: C
Explanation:

Let the no. be $xyz$ sum of digit is $(x+y+z)$ 

as $xyz=100x+10y+z$
then $xyz-(x+y+z)=99x+9y$  
$\therefore $ $\boxed{Always\, divisible\, by\, 9}$

The remainder when the polynomial $1+x^2+x^4+x^6+....+x^{22}$ is divided by $1+x+x^2+x^3+....+x^{11}$ is?

maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

  1. $0$

  2. $2$

  3. $1+x^2+x^4+...+x^{10}$

  4. $2(1+x^2+x^4+....+x^{10})$

Show answer
Correct Option: D
Explanation:
$ \left( \sum _{n=0}^N x^{n} \right) = \left ( \dfrac{x^{N+1}-1}{x-1} \right ) $

$ \Rightarrow Dividend = \left ( \dfrac{x^{24}-1}{x^{2}-1} \right ) $
$ Divisor = \left ( \dfrac{x^{12}-1}{x-1} \right ) $

Now,
$ \left ( \dfrac{x^{24}-1}{x^{2}-1} \right )  = \left ( \dfrac{x^{12}-1}{x-1} \right ) \left ( \dfrac{x^{12}-1+2}{x+1} \right ) = \left ( \dfrac{\left ( x^{12}-1 \right )^{2}}{x^{2}-1} \right ) + 2\left ( \dfrac{x^{12}-1}{x^{2}-1} \right ) $

$ \Rightarrow Remainder = 2\left ( 1+x^{2}+x^{4}...+x^{10} \right ) $

If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^7+x^5+x^3$ is divided by $(x^2+1)$, then the remainder is:

maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

  1. $1$

  2. $x^2+4$

  3. $-x$

  4. $x$

Show answer
Correct Option: C