Tag: remainder and factor theorems

Questions Related to remainder and factor theorems

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

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

  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. $p(x)=2x^2+2x+8; g(x)=x^2+x+9$;
    $q(x)=2; r(x)=-10$

Show answer
Correct Option: D
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

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 $r(x)$.

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

  1. $r(x)=7x^3+x^2-1$

  2. $r(x)=3x^2+2x+1$

  3. $r(x)=x-2$

  4. $r(x)=x+1$

Show answer
Correct Option: D
Explanation:

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

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

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

  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. $p(x)=x^2+x+3, g(x)=2x+3$;
    $q(x)=2x^2+x; r(x)=3x-2$

Show answer
Correct Option: C
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.

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

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

  1. $10-14x$

  2. $4x-10$

  3. $3x-5$

  4. $5-3x$

Show answer
Correct Option: A
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 $4x^2+3x-2$.

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

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

  1. Yes

  2. No

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
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 $x^5-4x^3+x^2+3x+1$

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

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

  1. $k=-2$ & $a=4$

  2. $k=5$ & $a=-5$

  3. $k=-3$ & $a=-7$

  4. None of these

Show answer
Correct Option: B
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, $k=5,a=-5$

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

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

  1. $15$

  2. $-15$

  3. $10$

  4. $-10$

Show answer
Correct Option: A
Explanation:

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

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

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

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

  1. $15x-14$

  2. $3x-14$

  3. $-15x+14$

  4. $-3x+14$

Show answer
Correct Option: C
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 $-(15x-14)=-15x+14$

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

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

  1. $\displaystyle 3x+1$

  2. $\displaystyle -3x-1$

  3. $\displaystyle -3x+1$

  4. $\displaystyle 3x-1$

Show answer
Correct Option: C
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 $-3x+1$.

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

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

  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. $\displaystyle x^{3}+4x^{2}-25x+78$

Show answer
Correct Option: B
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 $x^3-4x^2-25x+70$.