Fundamental Concepts of Algebra — Exercise 13(D)

Divide:

$-16ab^2c$ by $6abc$

Answer

Solving,

$\Rightarrow \dfrac{-16ab^2c}{6abc} = \dfrac{-8b}{3}$

Hence, $\bm{\dfrac{-16ab^2c}{6abc} = \dfrac{-8b}{3}}$.

Divide:

$25x^2y$ by $-5y^2$

Answer

Solving,

$\Rightarrow \dfrac{25x^2y}{-5y^2} = \dfrac{-5x^2}{y}$

Hence, $\bm{\dfrac{25x^2y}{-5y^2} = \dfrac{-5x^2}{y}}$.

Divide:

$8x + 24$ by 4

Answer

Solving,

$\Rightarrow \dfrac{8x + 24}{4}\\[1em] \Rightarrow \dfrac{8x}{4} + \dfrac{24}{4}\\[1em] \Rightarrow 2x + 6$

Hence, $\bm{\dfrac{8x + 24}{4} = 2x + 6}$.

Divide:

$4a^2 - a$ by $-a$

Answer

Solving,

$\Rightarrow \dfrac{4a^2 - a}{-a}\\[1em] \Rightarrow \dfrac{4a^2}{-a} - \dfrac{a}{-a}\\[1em] \Rightarrow -4a - (-1)\\[1em] \Rightarrow -4a + 1$

Hence, $\bm{\dfrac{4a^2 - a}{-a} = -4a + 1}$.

Divide:

$8m - 16$ by -8

Answer

Solving,

$\Rightarrow \dfrac{8m - 16}{-8}\\[1em] \Rightarrow \dfrac{8m}{-8} - \dfrac{16}{-8}\\[1em] \Rightarrow -m - (-2)\\[1em] \Rightarrow -m + 2$

Hence, $\bm{\dfrac{8m - 16}{-8} = -m + 2}$.

Divide:

$-50 + 40p$ by $10p$

Answer

Solving,

$\Rightarrow \dfrac{-50 + 40p}{10p}\\[1em] \Rightarrow \dfrac{-50}{10p} + \dfrac{40p}{10p}\\[1em] \Rightarrow -\dfrac{5}{p} + 4$

Hence, $\bm{\dfrac{-50 + 40p}{10p} = -\dfrac{5}{p} + 4}$.

Divide:

$4x^3 - 2x^2$ by $-x$

Answer

Solving,

$\Rightarrow \dfrac{4x^3 - 2x^2}{-x}\\[1em] \Rightarrow \dfrac{4x^3}{-x} - \dfrac{2x^2}{-x}\\[1em] \Rightarrow -4x^2 - (-2x)\\[1em] \Rightarrow -4x^2 + 2x$

Hence, $\bm{\dfrac{4x^3 - 2x^2}{-x} = -4x^2 + 2x}$.

Divide:

$10a^3 - 15a^2b$ by $- 5a^2$

Answer

Solving,

$\Rightarrow \dfrac{10a^3 - 15a^2b}{-5a^2}\\[1em] \Rightarrow \dfrac{10a^3}{-5a^2} - \dfrac{15a^2b}{-5a^2}\\[1em] \Rightarrow -2a - (-3b)\\[1em] \Rightarrow -2a + 3b$

Hence, $\bm{\dfrac{10a^3 - 15a^2b}{-5a^2} = -2a + 3b}$.

Divide:

$12x^3y - 8x^2y^2 + 4x^2y^3$ by $4xy$

Answer

Solving,

$\Rightarrow \dfrac{12x^3y - 8x^2y^2 + 4x^2y^3}{4xy}\\[1em] \Rightarrow \dfrac{12x^3y}{4xy} - \dfrac{8x^2y^2}{4xy} + \dfrac{4x^2y^3}{4xy}\\[1em] \Rightarrow 3x^2 - 2xy + xy^2$

Hence, $\bm{\dfrac{12x^3y - 8x^2y^2 + 4x^2y^3}{4xy} = 3x^2 - 2xy + xy^2}$.

Divide:

$9a^4b - 15a^3b^2 + 12a^2b^3$ by $- 3a^2b$

Answer

Solving,

$\Rightarrow \dfrac{9a^4b - 15a^3b^2 + 12a^2b^3}{-3a^2b}\\[1em] \Rightarrow \dfrac{9a^4b}{-3a^2b} - \dfrac{15a^3b^2}{-3a^2b} + \dfrac{12a^2b^3}{-3a^2b}\\[1em] \Rightarrow -3a^2 - (-5ab) + (-4b^2)\\[1em] \Rightarrow -3a^2 + 5ab - 4b^2$

Hence, $\bm{\dfrac{9a^4b - 15a^3b^2 + 12a^2b^3}{-3a^2b} = -3a^2 + 5ab - 4b^2}$.

Divide:

$n^2 - 2n + 1$ by $n - 1$

Answer

Solving,

$\begin{array}{l} n - 1\ \overline{\smash{\big)}\ n^2 - 2n + 1\ \smash{\big(}}\ n - 1 \\ \phantom{n - 1)}\underline{\underset{-}{}n^2 \underset{+}{-}\phantom{2}n\phantom{{} + 1}} \\ \phantom{n - 1\ \big)\ }\phantom{n^2\ {}}-n + 1 \\ \phantom{n --1))}\phantom{n^2\ {}}\underline{\underset{+}{-}n \underset{-}{+}1} \\ \phantom{n - 1}\phantom{++++}0 \end{array}$

Hence, $\bm{\dfrac{n^2 - 2n + 1}{n - 1} = n - 1}$.

Divide:

$m^2 - 2mn + n^2$ by $m - n$

Answer

Solving,

$\begin{array}{l} m - n\ \overline{\smash{\big)}\ m^2 - 2mn + n^2\ \smash{\big(}}\ m - n \\ \phantom{m - n)}\underline{\underset{-}{}m^2 \underset{+}{-}\phantom{2}mn\phantom{{} + n^2}} \\ \phantom{m - n\ \big)\ }\phantom{m^2\ {}}-mn + n^2 \\ \phantom{m - n)}\phantom{++(}\underline{\underset{+}{-}mn \underset{-}{+}n^2} \\ \phantom{m - n}\phantom{m^2 - 2mn + {}}0 \end{array}$

Hence, $\bm{\dfrac{m^2 - 2mn + n^2}{m - n} = m - n}$.

Divide:

$4a^2 + 4a + 1$ by $2a + 1$

Answer

Solving,

$\begin{array}{l} 2a + 1\ \overline{\smash{\big)}\ 4a^2 + 4a + 1\ \smash{\big(}}\ 2a + 1 \\ \phantom{2a + 1)}\underline{\underset{-}{}4a^2 \underset{-}{+}2a\phantom{{} + 1}} \\ \phantom{2a + 1\ \big)\ }\phantom{4a^2 + {}}2a + 1 \\ \phantom{2a + 1)}\phantom{4a^2 + {}}\underline{\underset{-}{}2a \underset{-}{+}1} \\ \phantom{2a + 1}\phantom{4a^2 + 2a + {}}0 \end{array}$

Hence, $\bm{\dfrac{4a^2 + 4a + 1}{2a + 1} = 2a + 1}$.

Divide:

$p^2 + 4p + 4$ by $p + 2$

Answer

Solving,

$\begin{array}{l} p + 2\ \overline{\smash{\big)}\ p^2 + 4p + 4\ \smash{\big(}}\ p + 2 \\ \phantom{p + 2)}\underline{\underset{-}{}p^2 \underset{-}{+}2p\phantom{{} + 4}} \\ \phantom{p + 2\ \big)\ }\phantom{p^2 + {}}2p + 4 \\ \phantom{p + 2)}\phantom{p^2 + {}}\underline{\underset{-}{}2p \underset{-}{+}4} \\ \phantom{p + 2}\phantom{p^2 + 2p + {}}0 \end{array}$

Hence, $\bm{\dfrac{p^2 + 4p + 4}{p + 2} = p + 2}$.

Divide:

$x^2 + 4xy + 4y^2$ by $x + 2y$

Answer

Solving,

$\begin{array}{l} x + 2y\ \overline{\smash{\big)}\ x^2 + 4xy + 4y^2\ \smash{\big(}}\ x + 2y \\ \phantom{x + 2y)}\underline{\underset{-}{}x^2 \underset{-}{+}2xy\phantom{{} + 4y^2}} \\ \phantom{x + 2y\ \big)\ }\phantom{x^2 + {}}2xy + 4y^2 \\ \phantom{x + 2y)}\phantom{x^2 + {}}\underline{\underset{-}{}2xy \underset{-}{+}4y^2} \\ \phantom{x + 2y}\phantom{x^2 + 2xy + {}}0 \end{array}$

Hence, $\bm{\dfrac{x^2 + 4xy + 4y^2}{x + 2y} = x + 2y}$.

Divide:

$2a^2 - 11a + 12$ by $a - 4$

Answer

Solving,

$\begin{array}{l} a - 4\ \overline{\smash{\big)}\ 2a^2 - 11a + 12\ \smash{\big(}}\ 2a - 3 \\ \phantom{a - 4)}\underline{\underset{-}{}2a^2 \underset{+}{-}\phantom{1}8a\phantom{{} + 12}} \\ \phantom{a - 4\ \big)\ }\phantom{2a^2\ {}}-3a + 12 \\ \phantom{a - 4)}\phantom{++()}\underline{\underset{+}{-}3a \underset{-}{+}12} \\ \phantom{a - 4}\phantom{2a^2 - 11a + {}}0 \end{array}$

Hence, $\bm{\dfrac{2a^2 - 11a + 12}{a - 4} = 2a - 3}$.

Divide:

$6x^2 + 5x - 6$ by $2x + 3$

Answer

Solving,

$\begin{array}{l} 2x + 3\ \overline{\smash{\big)}\ 6x^2 + 5x - 6\ \smash{\big(}}\ 3x - 2 \\ \phantom{2x + 3)}\underline{\underset{-}{}6x^2 \underset{-}{+}9x\phantom{{} - 6}} \\ \phantom{2x + 3\ \big)\ }\phantom{6x^2\ {}}-4x - 6 \\ \phantom{2x + 3)}\phantom{++()}\underline{\underset{+}{-}4x \underset{+}{-}6} \\ \phantom{2x + 3}\phantom{6x^2 + 5x - {}}0 \end{array}$

Hence, $\bm{\dfrac{6x^2 + 5x - 6}{2x + 3} = 3x - 2}$.

Divide:

$8a^2 + 4a - 60$ by $2a - 5$

Answer

Solving,

$\begin{array}{l} 2a - 5\ \overline{\smash{\big)}\ 8a^2 + \phantom{2}4a - 60\ \smash{\big(}}\ 4a + 12 \\ \phantom{2a - 5)}\underline{\underset{-}{}8a^2 \underset{+}{-}20a\phantom{{} - 60}} \\ \phantom{2a - 5\ \big)\ }\phantom{8a^2 + {}}24a - 60 \\ \phantom{2a - 5)}\phantom{8a^2 + {}}\underline{\underset{-}{}24a \underset{+}{-}60} \\ \phantom{2a - 5}\phantom{8a^2 + 24a - {}}0 \end{array}$

Hence, $\bm{\dfrac{8a^2 + 4a - 60}{2a - 5} = 4a + 12}$.

Divide:

$9x^2 - 24xy + 16y^2$ by $3x - 4y$

Answer

Solving,

$\begin{array}{l} 3x - 4y\ \overline{\smash{\big)}\ 9x^2 - 24xy + 16y^2\ \smash{\big(}}\ 3x - 4y \\ \phantom{3x - 4y)}\underline{\underset{-}{}9x^2 \underset{+}{-}12xy\phantom{{} + 16y^2}} \\ \phantom{3x - 4y\ \big)\ }\phantom{9x^2\ {}}-12xy + 16y^2 \\ \phantom{3x - 4y)}\phantom{++()}\underline{\underset{+}{-}12xy \underset{-}{+}16y^2} \\ \phantom{3x - 4y}\phantom{9x^2 - 24xy + {}}0 \end{array}$

Hence, $\bm{\dfrac{9x^2 - 24xy + 16y^2}{3x - 4y} = 3x - 4y}$.

Divide:

$15x^2 + 31xy + 14y^2$ by $5x + 7y$

Answer

Solving,

$\begin{array}{l} 5x + 7y\ \overline{\smash{\big)}\ 15x^2 + 31xy + 14y^2\ \smash{\big(}}\ 3x + 2y \\ \phantom{5x + 7y)}\underline{\underset{-}{}15x^2 \underset{-}{+}21xy\phantom{{} + 14y^2}} \\ \phantom{5x + 7y\ \big)\ }\phantom{15x^2 + {}}10xy + 14y^2 \\ \phantom{5x + 7y)}\phantom{15x^2 + {}}\underline{\underset{-}{}10xy \underset{-}{+}14y^2} \\ \phantom{5x + 7y}\phantom{15x^2 + 10xy + {}}0 \end{array}$

Hence, $\bm{\dfrac{15x^2 + 31xy + 14y^2}{5x + 7y} = 3x + 2y}$.

Divide:

$35a^3 + 3a^2b - 2ab^2$ by $5a - b$

Answer

Solving,

$\begin{array}{l} 5a - b\ \overline{\smash{\big)}\ 35a^3 + \phantom{1}3a^2b - 2ab^2\ \smash{\big(}}\ 7a^2 + 2ab \\ \phantom{5a - b)}\underline{\underset{-}{}35a^3 \underset{+}{-}\phantom{1}7a^2b\phantom{{} - 2ab^2}} \\ \phantom{5a - b\ \big)\ }\phantom{35a^3 + {}}10a^2b - 2ab^2 \\ \phantom{5a - b)}\phantom{35a^3 + {}}\underline{\underset{-}{}10a^2b \underset{+}{-}2ab^2} \\ \phantom{5a - b}\phantom{35a^3 + 10a^2b - {}}0 \end{array}$

Hence, $\bm{\dfrac{35a^3 + 3a^2b - 2ab^2}{5a - b} = 7a^2 + 2ab}$.

Divide:

$6x^3 + 5x^2 - 21x + 10$ by $3x - 2$

Answer

Solving,

$\begin{array}{l} 3x - 2\ \overline{\smash{\big)}\ 6x^3 + \phantom{1}5x^2 - 21x + 10\ \smash{\big(}}\ 2x^2 + 3x - 5 \\ \phantom{3x - 2)}\underline{\underset{-}{}6x^3 \underset{+}{-}\phantom{1}4x^2\phantom{{} - 21x + 10}} \\ \phantom{3x - 2\ \big)\ }\phantom{++())}9x^2 - 21x + 10 \\ \phantom{3x - 2)}\phantom{++())}\underline{\underset{-}{}9x^2 \underset{+}{-}\phantom{1}6x\phantom{{} + 10}} \\ \phantom{3x - 2\ \big)\ }\phantom{++++()}-15x + 10 \\ \phantom{3x - 2)}\phantom{+++++()}\underline{\underset{+}{-}15x \underset{-}{+}10} \\ \phantom{3x - 2}\phantom{++++++++()}0 \end{array}$

Hence, $\bm{\dfrac{6x^3 + 5x^2 - 21x + 10}{3x - 2} = 2x^2 + 3x - 5}$.

The area of a rectangle is $6x^2 - 4xy - 10y^2$ square unit and its length is $2x + 2y$ unit. Find its breadth.

Answer

Area of a rectangle = length × breadth

∴ Breadth = $\dfrac{\text{Area}}{\text{Length}} = \dfrac{6x^2 - 4xy - 10y^2}{2x + 2y}$

Dividing $6x^2 - 4xy - 10y^2$ by $2x + 2y$:

$\begin{array}{l} 2x + 2y\ \overline{\smash{\big)}\ 6x^2 - \phantom{1}4xy - 10y^2\ \smash{\big(}}\ 3x - 5y \\ \phantom{2x + 2y)}\underline{\underset{-}{}6x^2 \underset{-}{+}\phantom{1}6xy\phantom{{} - 10y^2}} \\ \phantom{2x + 2y\ \big)\ }\phantom{6x^2\ {}}-10xy - 10y^2 \\ \phantom{2x + 2y)}\phantom{++()}\underline{\underset{+}{-}10xy \underset{+}{-}10y^2} \\ \phantom{2x + 2y}\phantom{6x^2 - 10xy - {}}0 \end{array}$

Hence, the breadth of the rectangle is $\bm{(3x - 5y)}$ unit.

The area of a rectangular field is $25x^2 + 20xy + 3y^2$ square unit. If its length is $5x + 3y$ unit, find its breadth. Hence, find its perimeter.

Answer

Breadth = $\dfrac{\text{Area}}{\text{Length}} = \dfrac{25x^2 + 20xy + 3y^2}{5x + 3y}$

Dividing $25x^2 + 20xy + 3y^2$ by $5x + 3y$:

$\begin{array}{l} 5x + 3y\ \overline{\smash{\big)}\ 25x^2 + 20xy + 3y^2\ \smash{\big(}}\ 5x + y \\ \phantom{5x + 3y)}\underline{\underset{-}{}25x^2 \underset{-}{+}15xy\phantom{{} + 3y^2}} \\ \phantom{5x + 3y\ \big)\ }\phantom{25x^2 + {}}5xy + 3y^2 \\ \phantom{5x + 3y)}\phantom{25x^2 + {}}\underline{\underset{-}{}5xy \underset{-}{+}3y^2} \\ \phantom{5x + 3y}\phantom{25x^2 + 5xy + {}}0 \end{array}$

∴ Breadth = $5x + y$

$\text{ Perimeter } = 2 \times \text{ (length + breadth) }\\[1em] = 2 \times [(5x + 3y) + (5x + y)]\\[1em] = 2 \times (10x + 4y)\\[1em] = 20x + 8y$

Hence, the breadth = $\bm{(5x + y)}$ unit and the perimeter = $\bm{(20x + 8y)}$ unit.

Divide:

$2m^3n^5$ by $- mn$

Answer

Solving,

$\Rightarrow \dfrac{2m^3n^5}{-mn} \\[1em] \Rightarrow -2m^2n^4$

Hence, final result = $\bm{-2m^2n^4}$.

Divide:

$5x^2 - 3x$ by $x$

Answer

Solving,

$\Rightarrow \dfrac{5x^2 - 3x}{x}\\[1em] \Rightarrow \dfrac{5x^2}{x} - \dfrac{3x}{x}\\[1em] \Rightarrow 5x - 3$

Hence, final result = $\bm{5x - 3}$.

Divide:

$10x^3y - 9xy^2 - 4x^2y^2$ by $xy$

Answer

Solving,

$\Rightarrow \dfrac{10x^3y - 9xy^2 - 4x^2y^2}{xy}\\[1em] \Rightarrow \dfrac{10x^3y}{xy} - \dfrac{9xy^2}{xy} - \dfrac{4x^2y^2}{xy}\\[1em] \Rightarrow 10x^2 - 9y - 4xy$

Hence, final result = $\bm{10x^2 - 9y - 4xy}$.

Divide:

$3y^3 - 9ay^2 - 6ab^2y$ by $-3y$

Answer

Solving,

$\Rightarrow \dfrac{3y^3 - 9ay^2 - 6ab^2y}{-3y}\\[1em] \Rightarrow \dfrac{3y^3}{-3y} - \dfrac{9ay^2}{-3y} - \dfrac{6ab^2y}{-3y}\\[1em] \Rightarrow -y^2 - (-3ay) - (-2ab^2)\\[1em] \Rightarrow -y^2 + 3ay + 2ab^2$

Hence, final result = $\bm{-y^2 + 3ay + 2ab^2}$.

Divide:

$x^5 - 15x^4 - 10x^2$ by $- 5x^2$

Answer

Solving,

$\Rightarrow \dfrac{x^5 - 15x^4 - 10x^2}{-5x^2}\\[1em] \Rightarrow \dfrac{x^5}{-5x^2} - \dfrac{15x^4}{-5x^2} - \dfrac{10x^2}{-5x^2}\\[1em] \Rightarrow -\dfrac{1}{5}x^3 - (-3x^2) - (-2)\\[1em] \Rightarrow -\dfrac{1}{5}x^3 + 3x^2 + 2$

Hence, final result = $\bm{-\dfrac{1}{5}x^3 + 3x^2 + 2}$.

Divide:

$12a^2 + ax - 6x^2$ by $3a - 2x$

Answer

Solving,

$\begin{array}{l} 3a - 2x\ \overline{\smash{\big)}\ 12a^2 + \phantom{9}ax - 6x^2\ \smash{\big(}}\ 4a + 3x \\ \phantom{3a - 2x)}\underline{\underset{-}{}12a^2 \underset{+}{-}8ax\phantom{{} - 6x^2}} \\ \phantom{3a - 2x\ \big)\ }\phantom{12a^2 + {}}9ax - 6x^2 \\ \phantom{3a - 2x)}\phantom{12a^2 + {}}\underline{\underset{-}{}9ax \underset{+}{-}6x^2} \\ \phantom{3a - 2x}\phantom{12a^2 + 9ax - {}}0 \end{array}$

Hence, final result = $\bm{4a + 3x}$.

Divide:

$6x^2 - xy - 35y^2$ by $2x - 5y$

Answer

Solving,

$\begin{array}{l} 2x - 5y\ \overline{\smash{\big)}\ 6x^2 - \phantom{1}xy - 35y^2\ \smash{\big(}}\ 3x + 7y \\ \phantom{2x - 5y)}\underline{\underset{-}{}6x^2 \underset{+}{-}15xy\phantom{{} - 35y^2}} \\ \phantom{2x - 5y\ \big)\ }\phantom{6x^2 + {}}14xy - 35y^2 \\ \phantom{2x - 5y)}\phantom{6x^2 + {}}\underline{\underset{-}{}14xy \underset{+}{-}35y^2} \\ \phantom{2x - 5y}\phantom{6x^2 + 14xy - {}}0 \end{array}$

Hence, final result = $\bm{3x + 7y}$.

Divide:

$x^3 - 6x^2 + 11x - 6$ by $x^2 - 4x + 3$

Answer

Solving,

$\begin{array}{l} x^2 - 4x + 3\ \overline{\smash{\big)}\ x^3 - 6x^2 + 11x - 6\ \smash{\big(}}\ x - 2 \\ \phantom{x^2 - 4x + 3)}\underline{\underset{-}{}x^3 \underset{+}{-}4x^2 \underset{-}{+}\phantom{1}3x\phantom{{} - 6}} \\ \phantom{x^2 - 4x + 3\ \big)\ }\phantom{x^3\ {}}-2x^2 + \phantom{1}8x - 6 \\ \phantom{x^2 - 4x + 3)}\phantom{++}\underline{\underset{+}{-}2x^2 \underset{-}{+}\phantom{1}8x \underset{+}{-}6} \\ \phantom{x^2 - 4x + 3}\phantom{x^3 - 2x^2 + 8x - {}}0 \end{array}$

Hence, final result = $\bm{x - 2}$.

Divide:

$m^3 - 4m^2 + m + 6$ by $m^2 - m - 2$

Answer

Solving,

$\begin{array}{l} m^2 - m - 2\ \overline{\smash{\big)}\ m^3 - 4m^2 + \phantom{3}m + 6\ \smash{\big(}}\ m - 3 \\ \phantom{m^2 - m - 2)}\underline{\underset{-}{}m^3 \underset{+}{-}\phantom{4}m^2 \underset{+}{-}2m\phantom{{} + 6}} \\ \phantom{m^2 - m - 2\ \big)\ }\phantom{m^3\ {}}-3m^2 + 3m + 6 \\ \phantom{m^2 - m - 2)}\phantom{++()}\underline{\underset{+}{-}3m^2 \underset{-}{+}3m \underset{-}{+}6} \\ \phantom{m^2 - m - 2}\phantom{m^3 - 3m^2 + 3m + {}}0 \end{array}$

Hence, final result = $\bm{m - 3}$.