Tag: division algorithm for polynomials

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

Polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and "deg $q(x) = $ deg $ r(x)$", are

  1. $p(x)=2x^2+x; g(x)=2x^2-4$;
    $q(x)=2x-7; r(x)=-x+2$
  2. $p(x)=x^2+x-3; g(x)=x^2+x-1$;
    $q(x)=7; r(x)=-5$
  3. $p(x)=x^2+x; g(x)=x^2-4$;
    $q(x)=2x-1; r(x)=-x-2$
  4. $p(x)=2x^2+2x+8; g(x)=x^2+x+9$;
    $q(x)=2; r(x)=-10$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

according to division algorithm $p(x)=q(x)g(x)+r(x)$

degree of $q(x)$ is equal to $r(x)$ in all options.
only (D) option satisfies $p(x)=q(x)g(x)+r(x)$
$g(x)q(x)=2(x^2+x+9)=2x^2+2x+18=p(x)+10=p(x)-r(x)$ hence $p(x)=q(x)g(x)+r(x)$ in (D) satisfies division algorithm

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

On dividing $f(x)=2x^5+3x^4+4x^3+4x^2+3x+2$ by a polynomial $g(x)$, where $g(x)=x^3+x^2+x+1$, the quotient obtained as $2x^2+x+1$. Find the remainder $r(x)$.

  1. $r(x)=7x^3+x^2-1$
  2. $r(x)=3x^2+2x+1$
  3. $r(x)=x-2$
  4. $r(x)=x+1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

By remainder theorem,
$f(x)=q(x)g(x)+r(x)$
$\therefore 2x^5+3x^4+4x^3+4x^2+3x+2=(2x^2+x+1)(x^3+x^2+x+1)+r(x)$
$=2x^2(x^3+x^2+x+1)+x(x^3+x^2+x+1)+1(x^3+x^2+x+1)+r(x)$
$=2x^5+2x^4+2x^3+2x^2+x^4+x^3+x^2+x+x^3+x^2+x+1+r(x)$
$=2x^5+3x^4+4x^3+4x^2+2x+1+r(x)$
$r(x)=x+1$

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

Polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and "deg $p(x) = $ deg $q(x)$" are

  1. $p(x)=2x^2+2x+8, g(x)=4x+1$;
    $q(x)=x^2; r(x)=1$
  2. $p(x)=2x^2+2x+8, g(x)=5$;
    $q(x)=4; r(x)=4x-1$
  3. $p(x)=2x^2+2x+8, g(x)=2$;
    $q(x)=x^2+x+4; r(x)=0$
  4. $p(x)=x^2+x+3, g(x)=2x+3$;
    $q(x)=2x^2+x; r(x)=3x-2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

degree of $p(x)$ and $q(x)$ are equal in (A),(C),(D)

according to division algorithm, $p(x)=q(x)g(x)+r(x)$
in option (C), $g(x)q(x)=2(x^2+x+4)=2x^2+2x+8+0=g(x)q(x)+r(x)=p(x)$
hence option (C) is correct answer.

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

What should be added to $8x^4+14x^3-2x^2+7x-8$ so that the resulting polynomial is exactly divisible by $4x^2+3x-2$?

  1. $10-14x$
  2. $4x-10$
  3. $3x-5$
  4. $5-3x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$4x^2+3x-2)\overline {8x^4+14x^3-2x^2+7x-8}$ ( $2x^2+2x-1$
                          $\underline {\underset {-}{8}x^4\underset {-}{+}6x^3\underset {+}{-}4x^2}$
                                     $8x^3+2x^2+7x-8$
                                     $\underline {\underset {-}{8}x^3\underset {-}{+}6x^2\underset {+}{-}4x}$
                                             $-4x^2+11x-8$
                                             $\underline {\underset {+}{-}4x^2\underset {+}{-}3x\underset {-}{+}2}$
                                                          $14x-10$
We have to add $10-14x$ so that $8x^4+14x^3-2x^2+7x-8$ is completely divisible by $4x^2+3x-2$.

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

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm. $x^3-3x+1, x^5-4x^3+x^2+3x+1$

  1. Yes

  2. No

  3. Ambiguous

  4. Data insufficient
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$x^3-3x+1)\overline {x^5-4x^3+x^2+3x+1}$($x^2-1$
                         $\underset {-}{x^5}\underset {+}{-}3x^3\underset {-}{+}x^2$
                         $\overline {-x^3+3x+1}$
                         $\underline {\underset {+}{-}x^3\underset {-}{+}3x\underset {+}{-}1}$
                                              $2$
Since remainder is non-zero.
Therfore,$x^3-3x+1$ is not a factor of $x^5-4x^3+x^2+3x+1$

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

If the polynomial $f(x)=x^4-6x^3+16x^2-25x+10$ is divided by another polynomial $x^2-2x+k$, the remainder comes out to be $(x+a)$, then values of $k$ and $a$ are

  1. $k=-2$ & $a=4$
  2. $k=5$ & $a=-5$
  3. $k=-3$ & $a=-7$

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

$f(x)=$ is divided by another polynomial
$x^2-2x+k)\overline {x^4-6x^3+16x^2-25x+10}(x^2-4x+(8-k)$
                       $\underline {\underset {-}{x^4}\underset {+}{-2x^3}\underset{-}{+}kx^2}$
                       $-4x^3+(16-k)x^2-25x+10$
                       $\underline {\underset {-}{-4x^3}\underset {-}{+8x^2}                      \underset{+}{-}4kx}$
                       $(8-k)x^2+(4k-25)x+10$
                       $\underline {\underset {-}(8-k)x^2+\underset {-}(2k-16)x\underset{-}{+}(8k-k^2)}$
                       $(2k-9)x+(k^2-8k+10)$
But remainder is given $x+a$
$\therefore x+a=(2k-9)x+(k^2-8k+10)$
On equating coefficient, we get
$2k-9=1\Rightarrow k=5$
and $a=k^2-8k+10\Rightarrow a=25-40+10=-5$
Hence, $k=5,a=-5$

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

Find the value of $b$ for which the polynomial $2x^3+9x^2-x-b$ is exactly divisible by $2x+3$?

  1. $15$
  2. $-15$
  3. $10$
  4. $-10$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since $2x+3$ is a factor of the polynomial $p\left(x\right)=2x^3+9x^2-x-b$
Therefore, by Factor theorem $p\left(-\dfrac32\right)=0$
$\Rightarrow 2\left(-\dfrac32\right)^3+9\left(-\dfrac32\right)^2-\left(-\dfrac32\right)-b=0$

$\Rightarrow -\dfrac{27}4+\dfrac{81}4+\dfrac32-b=0$
$\Rightarrow \dfrac{-27+81+6}4-b=0\Rightarrow b=\dfrac{60}4=15$
$\therefore \space b=15$

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

What must be subtracted from or added to $8x^4+14x^3-2x^2+8x-12$ so that it may be exactly divisible by $4x^2+3x-2$?

  1. $15x-14$
  2. $3x-14$
  3. $-15x+14$
  4. $-3x+14$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$4x^2+3x-2)\overline {8x^4+14x^3-2x^2+8x-12}$ ( $2x^2+2x-1$
                            $\underline {\underset {-}{8}x^4\underset {-}{+}6x^3\underset {+}{-}4x^2}$
                            $8x^3+2x^2+8x-12$
                            $\underline {\underset {-}{8}x^3\underset {-}{+}6x^2\underset {+}{-}4x}$
                            $-4x^2+12x-12$
                            $\underline {\underset {+}{-}4x^2\underset {+}{-}3x\underset {-}{+}2}$
                                        $15x-14$

$\therefore$ The expression that must be subtracted is $15x-14$
and the expression that must be added is $-(15x-14)=-15x+14$

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

$\displaystyle \left( { 3x }^{ 2 }-x \right) \div \left( -x \right) $ is equal to

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

$\displaystyle \left( { 3x }^{ 2 }-x \right) \div \left( -x \right)$

By separating denominators, we get
$  =\dfrac { 3{ x }^{ 2 } }{ -x } +\dfrac { \left( -x \right)  }{ \left( -x \right)  } =-3x+1$
Hence, final result after given operation is $-3x+1$.

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

A polynomial when divided by $\displaystyle \left ( x-6 \right )$ gives a quotient $\displaystyle x^{2}+2x-13$ and leaves a remainder $-8$. Then polynomial is

  1. $\displaystyle x^{3}+4x^{2}+25x-78$
  2. $\displaystyle x^{3}-4x^{2}-25x+70$
  3. $\displaystyle x^{3}-4x^{2}-25x-70$
  4. $\displaystyle x^{3}+4x^{2}-25x+78$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $P$ be the polynomial. If $P$ is divided by $(x-6)$ then it leaves a remainder $-8$ and gives a quotient $x^2+2x-13$. Therefore, 

$\cfrac { P }{ x-6 } ={ x }^{ 2 }+2x-13-\cfrac { 8 }{ x-6 } \\ \Rightarrow P=(x-6)({ x }^{ 2 }+2x-13)-\frac { 8(x-6) }{ x-6 } \\ \Rightarrow P={ x }^{ 3 }+2{ x }^{ 2 }-13x-6{ x }^{ 2 }-12x+78-8\\ \Rightarrow P={ x }^{ 3 }-4{ x }^{ 2 }-25x+70$

Hence, the polynomial is $x^3-4x^2-25x+70$.