Tag: multiplication and division of algebraic expressions

Quizzes (4) Questions (13)

Questions Related to multiplication and division of algebraic expressions

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

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

  1. Divisible by 6

  2. Not divisible by 6

  3. Divisible by 9

  4. Not divisible by 9
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $ $\boxed{Always\, divisible\, by\, 9}$

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

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?

  1. $0$
  2. $2$
  3. $1+x^2+x^4+...+x^{10}$
  4. $2(1+x^2+x^4+....+x^{10})$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 ) $
Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

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:

  1. $1$
  2. $x^2+4$
  3. $-x$
  4. $x$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Divide the polynomial by x^2+1 by substituting x^2 = -1. The terms become powers of (-1), simplifying to -x.