Question 1: Divide:

i)  45x^7 \text{  by  } -9x^4      ii)  -60x^3 y^2 \text{  by  } -15xy

iii)  \displaystyle \frac{-3 }{ 4}  x^2 yz^3 \text{  by  } \frac{-2 }{ 3}  x^2 yz      iv)  63a^4 b^3 c^6 \text{  by  } -14a^2 b^5 c^4

Answer:

i)  \frac{45x^7 }{ -9x^4} =-5x^3

ii)  \frac{-60x^3 y^2 }{ -15xy} =4x^2 y

iii)  \frac{ \frac{-3 }{ 4} x^2 yz^3 } { \frac{-2 }{ 3} x^2 yz} = \frac{9 }{ 8} z^2

iv)  \frac{63a^4 b^3 c^6 }{ -14a^2 b^5 c^4} = \frac{-9 }{ 2} \frac{a^2 c^2 }{ b^2}

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Question 2: Divide:

i)  6ax-3cx+15x \text{  by  } -3x

ii)  10mn^2-15m^2 n^2+5m^3 n \text{  by  } -5mn

iii)  14x^3 y^4-7x^4 y^3-28x^3 y^6 \text{  by  } -7x^3 y^2

iv)  18a^6 b^3-30a^4 b^5+6a^4 b^4 \text{  by  } 6a^2 b^2

v)  8a^6 b-16a^2 b^2-6a^4 b^3 \text{  by  } -2a^2 b

vi)  \displaystyle \frac{ 1 }{ 2} p^2 q^3- \frac{5 }{ 3} p^3 q^2+\frac{1 }{ 4} p^3 q^3 \text{  by  } \frac{ -1 }{ 4} p^2 q^2

Answer:

i)  \displaystyle \frac{6ax-3cx+15x }{ -3x}  =-2a+c-5

ii)  \displaystyle \frac{10mn^2-15m^2 n^2+5m^3 n }{ -5mn}  =-2n+3mn-m^2

iii)  \displaystyle \frac{14x^3 y^4-7x^4 y^3-28x^3 y^6 }{ -7x^3 y^2}  =-2y^2+xy+4y^2

iv)  \displaystyle \frac{18a^6 b^3-30a^4 b^5+6a^4 b^4 }{ 6a^2 b^2}  =3a^4 b-5a^2 b^3+a^2 b^2

v)  \displaystyle \frac{8a^6 b-16a^2 b^2-6a^4 b^3 }{ -2a^2 b}  =-4a+8b+3a^2 b^2

vi)  \displaystyle \frac{ \frac{ 1 }{ 2} p^2 q^3- \frac{5 }{ 3} p^3 q^2+\frac{1 }{ 4} p^3 q^3 }{ \frac{ -1 }{ 4} p^2 q^2}  =-2q+\frac{5 }{ 2} p-pq

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Question 3:

i)  (x^2+8x+15 ) \text{  by  } (x+5 )      ii)  (4x^2+11x-3 ) \text{  by  } (x+3 )

iii)  (-2x^2+x+3 ) \text{  by  } (x+1 )      iv)  ( -2x^2+x+3 ) \text{  by  } (x+1 )

v)  ( x^3-3x^2 y+ 3xy^2-y^3 ) \text{  by  } (x-y )

Answer:

i)

\begin{array}{ r l l } x+5  & ) \overline{x^2+8x+15}( &  x+3   \\ (-) &  x^2 + 5 x &    \\ \hline & \hspace{1.0cm} 3x + 15 & \\ (-) &  \hspace{1.0cm} 3x + 15 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

ii)

\begin{array}{ r l l } x+3  & ) \overline{4x^2+11x-3}( &  4x-1   \\ (-) &  4x^2 + 12 x &    \\ \hline & \hspace{1.0cm} -x-3 & \\ (-) &  \hspace{1.0cm} -x-3 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

iii) 

\begin{array}{ r l l } x+1  & ) \overline{-2x^2+x+3}( &  -2x+3   \\ (-) &  -2x^2 - 2 x &    \\ \hline & \hspace{1.5cm} 3x+3 & \\ (-) &  \hspace{1.5cm} 3x+3 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

iv)

\begin{array}{ r l l } x+1 & ) \overline{x^3-9x^2+ 26x-24}( &  x^2-5x+6  \\ (-) &  x^3-4x^2 &    \\ \hline & \hspace{0.5cm} -5x^2+26x-24 & \\ (-) &  \hspace{0.5cm} -5x^2+20x & \\ \hline & \hspace{2.0cm} 6x-24 & \\ (-) &  \hspace{2.0cm} 6x-24 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

v)

\begin{array}{ r l l } x-y & ) \overline{x^3-3x^2 y+ 3xy^2-y^3}( &  x^2-2xy+y^2 \\ (-) &  x^3- x^2 y &    \\ \hline & \hspace{0.5cm} -2x^2 y+3xy^2-y^3 & \\ (-) &  \hspace{0.5cm} -2x^2 y+2xy^2 & \\ \hline & \hspace{2.0cm} xy^2-y^3 & \\ (-) &  \hspace{2.0cm} xy^2-y^3 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

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Question 4:

i)  (12x^2+11x+2 ) \text{  by  } (4x+1 )      ii)  (6x^2+x-15 ) \text{  by  } (2x-3 )

iii)  (6x^3-x^2- 26x-21 ) \text{  by  } (3x-7 )      iv)  ( 12x^2+7xy-12y^2 ) \text{  by  } (3x+4y )

Answer:

i)

\begin{array}{ r l l } 4x+1 & ) \overline{12x^2+11x+2}( &  3x+2  \\ (-) &  12x^2+ 3x &    \\ \hline & \hspace{1.0cm} 8x+2 & \\ (-) &  \hspace{1.0cm} 8x+2 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

ii)

\begin{array}{ r l l } 2x-3 & ) \overline{6x^2+x-15}( &  3x+5  \\ (-) &  6x^2- 9x &    \\ \hline & \hspace{1.0cm} 10x-15 & \\ (-) &  \hspace{1.0cm} 10x-15 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

iii)

\begin{array}{ r l l } 3x-7 & ) \overline{6x^3-x^2- 26x-21}( &  2 x^2+5x+3 \\ (-) &  6x^3-14 x^2 &    \\ \hline & \hspace{0.5cm} 15x^2-26x-21 & \\ (-) &  \hspace{0.5cm} 15x^2-35x & \\ \hline & \hspace{2.0cm} 9x-21 & \\ (-) &  \hspace{2.0cm} 9x-21 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

iv) 

\begin{array}{ r l l } 3x+4y & ) \overline{12x^2+7xy-12y^2}( &  4x-3y  \\ (-) &  12x^2+16xy &    \\ \hline & \hspace{1.5cm} -9xy-12y^2 & \\ (-) &  \hspace{1.5cm} -9xy-12y^2 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

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Question 5:

i)  (x^3-2x-1 ) \text{  by  } (x^2-x-1 )      ii)  (x^3-6x^2+11x-6 ) \text{  by  } (x^2-5x+6 )

iii)  (6x^5+4x^4-3x^3-1 ) \text{  by  } (3x^2-x+1 )      

iv)  ( 6x^5-28x^3+3x^2+30x-9 ) \text{  by  } (2x^2-6 )

Answer:

i)

\begin{array}{ r l l } x^2-x-1 & ) \overline{x^3-2x-1}( &  x+1 \\ (-) &  x^3- x^2-x &    \\ \hline & \hspace{1.0cm} x^2-x-1 & \\ (-) &  \hspace{1.0cm} x^2-x-1 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

ii)

\begin{array}{ r l l } x^2-5x+6 & ) \overline{x^3-6x^2+11x-6}( &  x-1 \\ (-) &  x^3- 5x^2+6x &    \\ \hline & \hspace{1.0cm} -x^2+5x-6 & \\ (-) &  \hspace{1.0cm} -x^2+5x-6 & \\ \hline & \hspace{1.75cm} 0 & \end{array}

iii)

\begin{array}{ r l l } 3x^2-x+1 & ) \overline{6x^5+4x^4-3x^3-1}( & 2x^3+2x^2-x-1 \\ (-) &  6x^5-2x^4+2x^3 &    \\ \hline & \hspace{0.5cm} 6x^4-5x^3-1 & \\ (-) &  \hspace{0.5cm} 6x^4-2x^3+2x^2 & \\ \hline & \hspace{1.2cm} -3x^3-2x^2-1 & \\ (-) &  \hspace{1.2cm} -3x^3+x^2-x & \\ \hline & \hspace{2.2cm} -3x^2+x-1 & \\ (-) &  \hspace{2.4cm} -3x^2+x-1 & \\ \hline & \hspace{3.2cm} 0 & \end{array}

iv) 

\begin{array}{ r l l } 2x^2-6 & ) \overline{6x^5-28x^3+3x^2+30x-9}( &  3x^3-5x+\frac{3}{2}  \\ (-) &  6x^5-18x^3 &    \\ \hline & \hspace{0.7cm} -10x^3+3x^2+30x-9 & \\ (-) &  \hspace{0.7cm} -10x^3 +30x & \\ \hline & \hspace{2.5cm} 3x^2 -9 & \\ (-) &  \hspace{2.5cm} 3x^2 -9 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

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Question 6:

i)  (x^3+27 ) \text{  by  } (x^2-3x+9 )      ii)  (x^4-81 ) \text{  by  } (x+3 )

iii)  (27x^3-8 ) \text{  by  } (3x-2 )      iv)  ( x^6-8 ) \text{  by  } (x^2-2 )

iv)  ( x^6-y^6 ) \text{  by  } (x-y )       iv)  ( 16x^4-81y^4 ) \text{  by  } (2x-3y )

Answer:

i)

\begin{array}{ r l l } x+3 & ) \overline{x^3+27\hspace{2.0cm}}( &  x^2-3x+9 \\ (-) &  x^3+3x^2 &    \\ \hline & \hspace{0.5cm} -3x^2+27 & \\ (-) &  \hspace{0.5cm} -3x^2-9x & \\ \hline & \hspace{2.0cm} 9x+27 & \\ (-) &  \hspace{2.0cm} 9x+27 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

ii)

\begin{array}{ r l l } x+3 & ) \overline{x^4-81\hspace{3.0cm}}( &  x^3-3x^2+9x-27 \\ (-) &  x^4+3x^3 &    \\ \hline & \hspace{0.5cm} -3x^3-81 & \\ (-) &  \hspace{0.6cm} -3x^3-9x^2 & \\ \hline & \hspace{2.0cm} 9x^2-81 & \\ (-) &  \hspace{2.0cm} 9x^2+27x & \\ \hline & \hspace{3.0cm} -27x-81 & \\ (-) &  \hspace{3.0cm} -27x-81 & \\ \hline & \hspace{3.75cm} 0 & \end{array}

iii)

\begin{array}{ r l l } 3x-2 & ) \overline{27x^3-8 \hspace{2.0cm} }( &  9x^2+6x+4 \\ (-) &  27x^3-18x^2 &    \\ \hline & \hspace{1.3cm} 18x^2-8 & \\ (-) &  \hspace{1.3cm} 18x^2-12x & \\ \hline & \hspace{2.6cm} 12x-8 & \\ (-) &  \hspace{2.6cm} 12x-8 & \\ \hline & \hspace{3.5cm} 0 & \end{array}

iv) 

\begin{array}{ r l l } x^2-2 & ) \overline{x^6-8 \hspace{2.0cm} }( &  x^4+2x^2-4 \\ (-) &  x^6-2x^4 &    \\ \hline & \hspace{0.9cm} 2x^4-8 & \\ (-) &  \hspace{0.9cm} 2x^4+4x^2 & \\ \hline & \hspace{1.6cm} -4x^2-8 & \\ (-) &  \hspace{1.6cm} -4x^2-8 & \\ \hline & \hspace{2.75cm} 0 & \end{array}

v)

\begin{array}{ r l l } x-y & ) \overline{x^6-y^6 \hspace{5.0cm} }( &  x^5+x^4 y+x^3 y^2+x^2 y^3+xy^4+y^5 \\ (-) &  x^6-x^5 y &    \\ \hline & \hspace{0.7cm} x^5 y-y^6 & \\ (-) &  \hspace{0.7cm} x^5 y-x^4 y^2 & \\ \hline & \hspace{1.8cm} x^4 y^2-y^6 & \\ (-) &  \hspace{1.8cm} x^4 y^2-x^3 y^3 & \\ \hline & \hspace{3.0cm} x^3 y^3-y^6 & \\ (-) &  \hspace{3.0cm} x^3 y^3-x^2 y^4 & \\ \hline & \hspace{4.0cm} x^2 y^4-y^6 & \\ (-) &  \hspace{4.0cm} x^2 y^4-xy^5 & \\ \hline & \hspace{5.2cm} xy^5-y^6 & \\ (-) &  \hspace{5.2cm} xy^5-y^6 & \\ \hline & \hspace{5.75cm} 0 & \end{array}

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Question 7:

i)  (6a^2-31a+47 ) \text{  is divided by  } (2a-5 )      

ii)  (2x^4-x^3+10x^2+8x-5 ) \text{  is divided by  } (x^2-x+6 )

iii)  (3t^5+7t^4-11t^3+8t^2-32t+5 ) \text{  is divided by  } (t^2+3t+2 )    

 iv)  ( x^6+3x^2+10 ) \text{  is divided by  } (x^3+1 )

Answer:

i)

\begin{array}{ r l l } 2a-5 & ) \overline{6a^2-31a+47}( &  x+3 \\ (-) &  \hspace{0.2cm} 6a^2-15a  &    \\ \hline & \hspace{1.0cm} -16a+47 & \\ (-) &  \hspace{1.0cm} -16a+40 & \\ \hline & \hspace{2.65cm} 7 & \end{array}

Quotient: x+3

Remainder: 7

ii)

\begin{array}{ r l l } x^2-x+6 & ) \overline{2x^4-x^3+10x^2+8x-5}( &  2x^2+x-1 \\ (-) &  \hspace{0.2cm} 2x^4-2x^3+12x^2 &    \\ \hline & \hspace{1.5cm} x^3-2x^2+8x-5 & \\ (-) &  \hspace{1.5cm} x^3-x^2+6x & \\ \hline & \hspace{2.0cm} -x^2+2x-5 & \\ (-) &  \hspace{2.0cm} -x^2+x-6 & \\ \hline & \hspace{3.0cm}  3x+1 & \end{array}

Quotient: 2x^2+x-1

Remainder: 3x+1

iii)

\begin{array}{ r l l } t^2+3t+2 & ) \overline{3t^5+7t^4-11t^3+8t^2-32t+5 }( &  3t^3-2t^2-11t+45 \\ (-) &  3t^5+9t^4+6t^3 &    \\ \hline & \hspace{0.8cm} -2t^4-17t^3+8t^2-32t+5 & \\ (-) &  \hspace{0.8cm} -2t^4-6t^3-4t^2 & \\ \hline & \hspace{1.7cm} - 11t^3+12t^2-32t+5 & \\ (-) &  \hspace{1.7cm} -11t^3-33t^2-22t & \\ \hline & \hspace{3.3cm} 45t^2-10t+5 & \\ (-) &  \hspace{3.4cm} 45t^2-135t+90 & \\ \hline & \hspace{4.8cm} 25t-85 & \end{array}

Quotient: 3t^3-2t^2-11t+45

Remainder: 25t-85

iv) 

\begin{array}{ r l l } x^3+1 & ) \overline{x^6+3x^2+10 }( &  x^3-1 \\ (-) &  x^6+x^3 &    \\ \hline & \hspace{0.9cm} -x^3+3x^2+10 & \\ (-) &  \hspace{0.9cm} -x^3-1 & \\ \hline & \hspace{2.00cm} 3x^2+11 & \end{array}

Quotient: x^3-1

Remainder: 3x^2+11

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Question 8: Show by division method that (2a^2-a+3) is a factor of (6a^5-a^4+4a^3-5a^2-9-15)

Answer:

\begin{array}{ r l l } 2a^2-a+3 & ) \overline{6a^5-a^4+4a^3-5a^2-a-15} ( & 3a^3+a^2-2a-5 \\ (-) & 6a^5-3a^4+9a^3 &  \\ \hline  & \hspace{1.0cm}2a^4-5a^3-5a^2-a-15 &  \\ (-) & \hspace{1.0cm}2a^4-a^3+3a^2 &  \\ \hline  & \hspace{2.0cm}-4a^3-8a^2-a-15 &  \\ (-) & \hspace{2.0cm}-4a^3+2a^2-6a &  \\ \hline & \hspace{3.0cm}-10a^2+5a-15 &  \\ (-) & \hspace{3.0cm}-10a^2+5a-15 &  \\ \hline &\hspace{4.0cm} 0 & \end{array}

Quotient: 3a^3+a^2-2a-5

Remainder: 0

Hence it is a factor.

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Question 9: What must be subtracted from 8x^4+14x^3-2x^4+7x-8 so that the resulting polynomial is exactly divisible by (4x^2+3x-2) .

Answer:

\begin{array}{ r l l } 4x^2+3x-2 & ) \overline{8x^4+14x^3-2x^2+7x-8} ( & 2x^2+2x-1 \\ (-) & 8x^4+6x^3-4x^2 & \\ \hline & \hspace{1.0cm} 8x^3+2x^2+7x-8 & \\ (-) & \hspace{1.0cm} 8x^3+6x^2-4x & \\ \hline  & \hspace{2.0cm}-4x^2+11x-8 & \\ (-) & \hspace{2.0cm}-4x^2-3x+2 & \\ \hline  & \hspace{3.0cm}14x-10 & \\ \end{array}

Quotient: 2x^2+2x-1

Remainder: 14x-10

Therefore subtract 14x-10

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Question 10: If (2x-3y) units, (7x+y) units, (x+12y) units of (3y-4x) units are the lengths of the side at a quadrilateral, find the perimeter of the quadrilateral.

Answer:

Perimeter  = (2x-3y)+(7x+y)+(x+12y)+(-4x+3y) = 6x+13y

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Question 11: There are (3x+5) she has in a library and in each shelf there are (5x+3y) books. How many books are there in the library?

Answer:

Total No. of Book = (3x+5y)(5x+3y) = 15x^2+25xy+9xy+15y^2 = 15x^2+34xy+15y^2

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Question 12: The length and breadth of a rectangle are (a+5b) units and (7a-b) units respectively. The perimeter of the rectangle is equal to the perimeter of the square. Find how much in the area of the rectangle can there that of the square.

Answer:

Perimeter of Rectangle =2(a+5b)+2(7a-b) = 16a+8b

Area of rectangle = (a+5b)(7a-b) = 7a^2+35ab-ab-5b^2 = 7a^2+34ab-5b^2

Side of the square \displaystyle = \frac{16a+8b}{4}  =(4a+2b)

Area of square = (4a+2b)(4a+2b) = 16a^2+8ab+8ab+4b^2 = 16a^2+16ab+4ab^2

Area of square – Area of rectangle = 16a^2+16ab+4b^2-7a^2-34ab+5b^2 = 9a^2-18ab+9b^2

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Question 13: If a sum of Rs. (16x^3-46x^2+39x-9) in to be divide a equally among (8x-3) persons, find the amount received by each person.

Answer:

\begin{array}{ r l l } 8x-3 & )\overline{16x^3-46x^2+39x-9}( & 2x^2-5x+3  \\ (-) & 16x^3-6x^2 &   \\ \hline  & \hspace{1.0cm} -40x^2+39x-9 &   \\ (-) & \hspace{1.0cm} -40x^2+15x &   \\ \hline  & \hspace{2.0cm} 24x-3 &   \\ (-) & \hspace{2.0cm} 24x-3 &   \\ \hline  & \hspace{3.0cm} 0 & \end{array}

Quotient: 2x^2-5x+3

Remainder: 0

Each one will get 2x^2-5x+3 Rs.

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Question 14: The product of two numbers in (x^6-y^6) If one of the numbers in (x-y) then find the other number.

Answer:

\begin{array}{ r l l } x-y & )\overline{x^6-y^6 \hspace{5.0cm} } ( & x^5+x^4 y+x^3 y^2+x^2 y^3+xy^4+y^5  \\ (-) & x^6-x^5 y &   \\ \hline  & \hspace{1.0cm} x^5 y-y^6 &   \\ (-) & \hspace{1.0cm} x^5 y-x^4 y^2 &   \\ \hline  & \hspace{2.0cm} x^4 y^2-y^6 &   \\ (-) & \hspace{2.0cm} x^4 y^2-x^3 y^3 &   \\ \hline  & \hspace{3.0cm} x^3 y^3-y^6 &   \\ (-) & \hspace{3.0cm} x^3 y^3-x^2 y^4 &   \\ \hline  & \hspace{4.0cm} x^2 y^4-y^6 &   \\ (-) & \hspace{4.0cm} x^2 y^4-xy^5 &   \\ \hline   & \hspace{5.0cm} xy^5-y^6 &   \\ (-) & \hspace{5.0cm} xy^5-y^6 &   \\ \hline  & \hspace{6.0cm} 0 & \end{array}

The other number =x^5+x^4 y+x^3 y^2+x^2 y^3+xy^4+y^5

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Question 15: Divide 6x^4-13x^3+2x^2+22x-24 by the product of (2x-3) and (x^2-2x+2)

Answer:

Step 1:

\begin{array}{ r l l } x^2-2x+2 & )\overline{6x^4-13x^3+2x^2+22x-24}( & 6x^2-x-12  \\ (-) & 6x^4-12x^3+12x^2 &   \\ \hline  & \hspace{1.0cm} -x^3-10x^2+22x-24 &   \\ (-) & \hspace{1.0cm} -x^3+2x^2-2x &   \\ \hline  & \hspace{2.0cm}  -12x^2+24x-24 &   \\ (-) & \hspace{2.0cm} -12x^2+24x-24 &   \\ \hline  & \hspace{3.0cm} 0 & \end{array}

Step 2:

\begin{array}{ r l l } 2x-3 & )\overline{5x^2-x-12}( & 3x+4  \\ (-) & 6x^2-9x &   \\ \hline  & \hspace{1.0cm} 8x-12 &   \\ (-) & \hspace{1.0cm} 8x-12 &   \\ \hline  & \hspace{1.5cm} 0 & \end{array}