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)  $63a^4 b^3 c^6 \text{ by } -14a^2 b^5 c^4$

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

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

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)  $( 12x^2+7xy-12y^2 ) \text{ by } (3x+4y )$

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

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

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

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 $(6a^5-a^4+4a^3-5a^2-9-15)$

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

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 $(4x^2+3x-2)$.

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

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.

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?

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.

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.

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

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.

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

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 $(x^2-2x+2)$

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:

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