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 remainder obtained when the polynomial $1+x+x^ {3}+x^ {9}+x^ {27}+x^ {81}+x^ {243}$ is divisible by $x-1$ is

  1. $3$
  2. $5$
  3. $7$
  4. $11$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$p{\left( x \right)} = 1 + x + {x}^{3} + {x}^{9} + {x}^{27} + {x}^{81} + {x}^{243}$

Let $q{\left( x \right)}$ be the quotient when $P{\left( x \right)}$ divided by $\left( x - 1 \right)$.
Therefore,
$P{\left( x \right)} = \left( x - 1 \right) \cdot q{\left( x \right)} + A$
$P{\left( 1 \right)} = \left( 1 - 1 \right) \cdot q {\left( x \right)} + A$
$7 = 0 + A$
$A = 7$
Hence the remainder is $7$.

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 $A=2x^{3}+5x^{2}+4x+1$ and $B=2x^{2}+3x+1$, then find the quotient from the following four option, when A is divided by B.

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


$\dfrac{A}{B} = \dfrac{2x^{3}+5x^{2}+4x+1}{2x^{2}+3x+1}$

$=\dfrac{2x^{3}+(3x^{2}+2x^{2})+(x+3x)+1}{2x^{2}+2x+1}$

$=\dfrac{(2x^{3}+3x^{2}+x)+(2x^{2}+3x+1)}{2x^{2}+3x+1}$

$=\dfrac{(2x^{2}+3x+1)(x+1)}{2x^{2}+3x+1}=x+1$

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

Evaluate: $\displaystyle \frac{a^3\, +\, b^3\, +\, c^3\, -\, 3abc}{a^2\, +\, b^2\, +\, c^2\, -\, ab\, -\, bc\, -\, ca}$

  1. $0$
  2. $a + b + c$
  3. $1$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\cfrac { { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc }{ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca } \ =\cfrac { \left( a+b+c \right) \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca \right)  }{ { (a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca) } \ =a+b+c$.

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 x + 2 and x-1 are the factors of $x^3 + 10x^2+mx + n$, then the values of m and n are respectively

  1. 5 and -3

  2. 17 and -8

  3. 7 and-18

  4. 23 and -19
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Here, $x + 2$ is a factor of $x^3 + 10x^2 + mx + n$
$x =-2$
$(-2)^3 + 10(-2)^2 + m(-2) + n =0$
$ -8 + 40 = 2m - n $
$2m -n = 32$                     .....(i)
Again, $x-1 $ is a factor of $x^3 + 10x^2 + mx + n$
$x =1$
$1+10+m+n=0$
$m + n =-11$                    .....(ii)
Adding (i) and (ii). we get,
$3m = 21$
$m=7$
By putting m in (i). we get,
$2(7) - n = 32 $
$ - n = 18 $
   $n = -18$
Option C is correct.

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 expression $2x^3 + ax^2 + bx +3$, where a and b are constants, has a factor of x-1 and leaves a remainder of 15 when divided by x+2. Find the value of a and b respectively.

  1. $-3, 8$
  2. $3,-8$
  3. $-3,-8$
  4. $3, 8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

f(x)=$ 2x^3+ax^2-bx+3$ 
At x=2 
f(2)=15 
f(1)=0 
f(x)=$ 2x^3+ax^2-bx+3$ 
f(1)= 2+a-b+3=0 
a-b+5=0......A 
f(x)= $2x^3+ax^2-bx+3$ 
f(2)=$ 2(2^3)+a(2^2)-2b+3=15 $
4a-2b=-4 
Multiply A by 2 and subtract from above equation 
4a-2b=-4 
2a-2b+10=0 
2a-10=-4 
2a= 6 
a=3 
From A 
3-b+5=0 
8-b=0 
b=8 
So a=3 and b=8

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 $(x^{3} + 5x^{2} + 10k)$ leaves remainder $-2x$ when divided by $(x^{2} + 2)$, then what is the value of k?

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

$x^{3} + 5x^{2} + 10k$
$= (x^{2} + 2)(x + 5) + 10k - 2x - 10$
$\Rightarrow 10k - 2x - 10 = -2x$
$\Rightarrow 10k - 10 = 0$ or $k = 1$.

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 polynomial $f(x)={ x }^{ 4 }-2{ x }^{ 3 }+3{ x }^{ 2 }-ax+b$ when divided by $(x-1)$ and $(x+1)$ leaves the remainders $5$ and $19$ respectively. Find the values of $a$ and $b$. Hence, find the remainder when $f(x)$ is divided by $(x-2)$

  1. $a=6,b=8$ and $remainder=10$
  2. $a=5,b=8$ and $remainder=10$
  3. $a=5,b=7$ and $remainder=10$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

    by remainder theorem,

      $f(1)=5$ $and$ $f(-1)=14$

     $\therefore 1-2+3-a+b=5\Rightarrow b-a=3$  and
     $1+2+3+a+b=19\Rightarrow a+b=13$ 
      $b=8$ $a=5$ 
       and remainder when $f(x)$ is divided by $(x-2)$ is
         $f(2)=16-16+12-10+8=10$

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

$32x^{10}-33x^{5}+1$ is divisible by 

  1. $x-1$
  2. $x-2$
  3. $x-3$
  4. $x-4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$32x^{10}- 33 x^5 +1=0$
Let $m = x^5$
$\Rightarrow 32 m^2 - 33 m + 1 =0$
$\Rightarrow (32 m -1)(m-1)=0$
$\Rightarrow (32x^5-1)(x^5-1)=0$
$\therefore 32x^{10}- 33 x^5 +1=(32x^5-1)(x^5-1)$

$x^n-y^n$ is always divisible by $(x-y)$

$\therefore \,32x^{10}- 33 x^5 +1$ is divisible by $(x-1)$

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 $({ 15 }^{ 23 }+{ 23 }^{ 23 })$ is divided by $19$, is 

  1. $4$
  2. $17$
  3. $23$
  4. $0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given, $(15)^{23}+(23)^{23}$

$=(19-4)^{23}+(19+4)^{23}$

$\Rightarrow $ In bino  expansion  of above expression the term containing  19 well be cancelled 
as they disable by 19 then remaining term are 

$\Rightarrow  (-4)^{23}+(4)^{23}=0$

Therefore the remainder is exactly  zero .
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 $(x^{100} + 2x^{99} + K)$ is exactly divisible by $(x + 1)$, find the value of 'K'

  1. $1$
  2. $2$
  3. $-2$
  4. $-3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x^{100}+2x^{99}+k$ is exactly divisible by $(x+1)$

$\therefore x=-1$ is the root of $x^{100}+2x^{99}+k$
$\Rightarrow (-1)^{100}+2(-1)^{99}+k=0$ 
$\Rightarrow 1-2+k=0$ 
$\Rightarrow \boxed{k=1}$