Fundamental Concepts of Algebra — Exercise 13(B)

Fill in the blanks:

(i) $8x + 5x =$ .............

(ii) $8x - 5x =$ .............

(iii) $6xy^2 + 9xy^2 =$ .............

(iv) $6xy^2 - 9xy^2 =$ .............

(v) The sum of $8a, 6a$ and $5b =$ .....................

(vi) The addition of $5, 7xy, 6$ and $3xy =$ ....................

(vii) $4a + 3b - 7a + 4b =$ ....................

(viii) $- 15x + 13x + 8 =$ ....................

(ix) $6x^2y + 13xy^2 - 4x^2y + 2xy^2 =$ ....................

(x) $16x^2 - 9x^2 =$ .................... and $25xy^2 - 17xy^2 =$ ....................

Answer

(i) $8x + 5x = (8 + 5)x = \bm{13x}$.

(ii) $8x - 5x = (8 - 5)x = \bm{3x}$.

(iii) $6xy^2 + 9xy^2 = (6 + 9)xy^2 = \bm{15xy^2}$.

(iv) $6xy^2 - 9xy^2 = (6 - 9)xy^2 = \bm{-3xy^2}$.

(v) $8a + 6a + 5b = (8 + 6)a + 5b = \bm{14a + 5b}$.

(vi) $5 + 7xy + 6 + 3xy = (5 + 6) + (7 + 3)xy = \bm{11 + 10xy}$.

(vii) $4a + 3b - 7a + 4b = (4 - 7)a + (3 + 4)b = \bm{-3a + 7b}$.

(viii) $-15x + 13x + 8 = (-15 + 13)x + 8 = \bm{-2x + 8}$.

(ix) $6x^2y + 13xy^2 - 4x^2y + 2xy^2 = (6 - 4)x^2y + (13 + 2)xy^2 = \bm{2x^2y + 15xy^2}$.

(x) $16x^2 - 9x^2 = \bm{7x^2}$ and $25xy^2 - 17xy^2 = \bm{8xy^2}$.

Add:

$-9x, 3x$ and $4x$

Answer

Solving,

$\Rightarrow -9x + 3x + 4x \\[1em] \Rightarrow (-9 + 3 + 4)x \\[1em] \Rightarrow -2x$

Hence, the required sum = $\bm{-2x}$.

Add:

$23y^2, 8y^2$ and $-12y^2$

Answer

Solving,

$\Rightarrow 23y^2 + 8y^2 + (-12y^2) \\[1em] \Rightarrow (23 + 8 - 12)y^2 \\[1em] \Rightarrow 19y^2$

Hence, the required sum = $\bm{19y^2}$.

Add:

$18pq, -15pq$ and $3pq$

Answer

Solving,

$\Rightarrow 18pq + (-15pq) + 3pq \\[1em] \Rightarrow (18 - 15 + 3)pq \\[1em] \Rightarrow 6pq$

Hence, the required sum = $\bm{6pq}$.

Simplify:

$3m + 12m - 5m$

Answer

Solving,

$\Rightarrow 3m + 12m - 5m \\[1em] \Rightarrow (3 + 12 - 5)m \\[1em] \Rightarrow 10m$

Hence, the simplified expression = $\bm{10m}$.

Simplify:

$7n^2 - 9n^2 + 3n^2$

Answer

Solving,

$\Rightarrow 7n^2 - 9n^2 + 3n^2 \\[1em] \Rightarrow (7 - 9 + 3)n^2 \\[1em] \Rightarrow n^2$

Hence, the simplified expression = $\bm{n^2}$.

Simplify:

$25zy - 8zy - 6zy$

Answer

Solving,

$\Rightarrow 25zy - 8zy - 6zy \\[1em] \Rightarrow (25 - 8 - 6)zy \\[1em] \Rightarrow 11yz$

Hence, the simplified expression = $\bm{11yz}$.

Simplify:

$-5ax^2 + 7ax^2 - 12ax^2$

Answer

Solving,

$\Rightarrow -5ax^2 + 7ax^2 - 12ax^2 \\[1em] \Rightarrow (-5 + 7 - 12)ax^2 \\[1em] \Rightarrow -10ax^2$

Hence, the simplified expression = $\bm{-10ax^2}$.

Simplify:

$-16am + 4mx + 4am - 15mx + 5am$

Answer

Solving,

$\Rightarrow -16am + 4mx + 4am - 15mx + 5am \\[1em] \Rightarrow (-16 + 4 + 5)am + (4 - 15)mx \\[1em] \Rightarrow -7am - 11mx$

Hence, the simplified expression = $\bm{-7am - 11mx}$.

Add:

$a + b$ and $2a + 3b$

Answer

Solving,

$\begin{array}{rrr} & a & + b \\ + & 2a & + 3b \\ \hline & 3a & + 4b \\ \end{array}$

Hence, the required sum = $\bm{3a + 4b}$.

Add:

$2x + y$ and $3x - 4y$

Answer

Solving,

$\begin{array}{rrr} & 2x & + y \\ + & 3x & - 4y \\ \hline & 5x & - 3y \\ \end{array}$

Hence, the required sum = $\bm{5x - 3y}$.

Add:

$-3a + 2b$ and $3a + b$

Answer

Solving,

$\begin{array}{rrr} & -3a & + 2b \\ + & 3a & + b \\ \hline & & 3b \\ \end{array}$

Hence, the required sum = $\bm{3b}$.

Add:

$4 + x, 5 - 2x$ and $6x$

Answer

Solving,

$\begin{array}{rrr} & 4 & + x \\ & 5 & - 2x \\ + & & 6x \\ \hline & 9 & + 5x \\ \end{array}$

Hence, the required sum = $\bm{9 + 5x}$.

Find the sum of:

$3x + 8y + 7z, 6y + 4z - 2x$ and $3y - 4x + 6z$

Answer

Solving,

$\begin{array}{rrrr} & 3x & + 8y & + 7z \\ & -2x & + 6y & + 4z \\ + & -4x & + 3y & + 6z \\ \hline & -3x & + 17y & + 17z \\ \end{array}$

Hence, the required sum = $\bm{-3x + 17y + 17z}$.

Find the sum of:

$3a + 5b + 2c, 2a + 3b - c$ and $a + b + c$

Answer

Solving,

$\begin{array}{rrrr} & 3a & + 5b & + 2c \\ & 2a & + 3b & - c \\ + & a & + b & + c \\ \hline & 6a & + 9b & + 2c \\ \end{array}$

Hence, the required sum = $\bm{6a + 9b + 2c}$.

Find the sum of:

$4x^2 + 8xy - 2y^2$ and $8xy - 5y^2 + x^2$

Answer

Solving,

$\begin{array}{rrrr} & 4x^2 & + 8xy & - 2y^2 \\ + & x^2 & + 8xy & - 5y^2 \\ \hline & 5x^2 & + 16xy & - 7y^2 \\ \end{array}$

Hence, the required sum = $\bm{5x^2 + 16xy - 7y^2}$.

Find the sum of:

$9x^2 - 6x + 7, 5 - 4x$ and $6 - 3x^2$

Answer

Solving,

$\begin{array}{rrrr} & 9x^2 & - 6x & + 7 \\ & & -4x & + 5 \\ + & -3x^2 & & + 6 \\ \hline & 6x^2 & - 10x & + 18 \\ \end{array}$

Hence, the required sum = $\bm{6x^2 - 10x + 18}$.

Find the sum of:

$5x^2 - 2xy + 3y^2, - 2x^2 + 5xy + 9y^2$ and $3x^2 - xy - 4y^2$

Answer

Solving,

$\begin{array}{rrrr} & 5x^2 & - 2xy & + 3y^2 \\ & -2x^2 & + 5xy & + 9y^2 \\ + & 3x^2 & - xy & - 4y^2 \\ \hline & 6x^2 & + 2xy & + 8y^2 \\ \end{array}$

Hence, the required sum = $\bm{6x^2 + 2xy + 8y^2}$.

Find the sum of:

$a^2 + b^2 + 2ab, 2b^2 + c^2 + 2bc$ and $4c^2 - a^2 + 2ac$

Answer

Solving,

$\begin{array}{rrrrrrr} & a^2 & + b^2 & & + 2ab & & \\ & & 2b^2 & + c^2 & & + 2bc & \\ + & -a^2 & & + 4c^2 & & & + 2ac \\ \hline & & 3b^2 & + 5c^2 & + 2ab & + 2bc & + 2ac \\ \end{array}$

Hence, the required sum = $\bm{3b^2 + 5c^2 + 2ab + 2bc + 2ac}$.

Find the sum of:

$9ax - 6bx + 8, 4ax + 8bx - 7$ and $- 6ax - 4bx - 3$

Answer

Solving,

$\begin{array}{rrrr} & 9ax & - 6bx & + 8 \\ & 4ax & + 8bx & - 7 \\ + & -6ax & - 4bx & - 3 \\ \hline & 7ax & - 2bx & - 2 \\ \end{array}$

Hence, the required sum = $\bm{7ax - 2bx - 2}$.

Find the sum of:

$abc + 2ba + 3ac, 4ca - 4ab + 2bca$ and $2ab - 3abc - 6ac$

Answer

Solving,

$\begin{array}{rrrr} & abc & + 2ab & + 3ac \\ & 2abc & - 4ab & + 4ac \\ + & -3abc & + 2ab & - 6ac \\ \hline & & & ac \\ \end{array}$

Hence, the required sum = $\bm{ac}$.

Find the sum of:

$4a^2 + 5b^2 - 6ab, 3ab, 6a^2 - 2b^2$ and $4b^2 - 5ab$

Answer

Solving,

$\begin{array}{rrrr} & 4a^2 & + 5b^2 & - 6ab \\ & & & 3ab \\ & 6a^2 & - 2b^2 & \\ + & & 4b^2 & - 5ab \\ \hline & 10a^2 & + 7b^2 & - 8ab \\ \end{array}$

Hence, the required sum = $\bm{10a^2 + 7b^2 - 8ab}$.

Find the sum of:

$x^2 + x - 2, 2x - 3x^2 + 5$ and $2x^2 - 5x + 7$

Answer

Solving,

$\begin{array}{rrrr} & x^2 & + x & - 2 \\ & -3x^2 & + 2x & + 5 \\ + & 2x^2 & - 5x & + 7 \\ \hline & & -2x & + 10 \\ \end{array}$

Hence, the required sum = $\bm{-2x + 10}$.

Find the sum of:

$4x^3 + 2x^2 - x + 1, 2x^3 - 5x^2 - 3x + 6, x^2 + 8$ and $5x^3 - 7x$

Answer

Solving,

$\begin{array}{rrrrr} & 4x^3 & + 2x^2 & - x & + 1 \\ & 2x^3 & - 5x^2 & - 3x & + 6 \\ & & x^2 & & + 8 \\ + & 5x^3 & & - 7x & \\ \hline & 11x^3 & - 2x^2 & - 11x & + 15 \\ \end{array}$

Hence, the required sum = $\bm{11x^3 - 2x^2 - 11x + 15}$.

Find the sum of:

$x$ and $3y$

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{x + 3y}$.

Find the sum of:

$-2a$ and +5

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{-2a + 5}$.

Find the sum of:

$-4x^2$ and $+7x$

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{-4x^2 + 7x}$.

Find the sum of:

$+4a$ and $-7b$

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{4a - 7b}$.

Find the sum of:

$x^3, 3x^2y$ and $2y^2$

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{x^3 + 3x^2y + 2y^2}$ .

Find the sum of:

11 and $-by$

Answer

Since the given terms are unlike terms, their sum cannot be simplified further.

Sum = $\bm{11 - by}$.

The sides of a triangle are $2x + 3y, x + 5y$ and $7x - 2y$. Find its perimeter.

Answer

Perimeter of a triangle = sum of its three sides

Solving,

$\Rightarrow (2x + 3y) + (x + 5y) + (7x - 2y) \\[1em] \Rightarrow (2 + 1 + 7)x + (3 + 5 - 2)y \\[1em] \Rightarrow 10x + 6y$

Hence, the perimeter of the triangle = $\bm{10x + 6y}$.

The two adjacent sides of a rectangle are $6a + 9b$ and $8a - 4b$. Find its perimeter.

Answer

Perimeter of a rectangle = 2 × (sum of two adjacent sides)

Solving,

$\Rightarrow 2 \times [(6a + 9b) + (8a - 4b)] \\[1em] \Rightarrow 2 \times (14a + 5b) \\[1em] \Rightarrow 28a + 10b$

Hence, the perimeter of the rectangle = $\bm{28a + 10b}$.

Subtract the second expression from the first:

$2a + b, a + b$

Answer

Solving,

$\Rightarrow (2a + b) - (a + b) \\[1em] \Rightarrow 2a + b - a - b \\[1em] \Rightarrow 2a - a + b - b \\[1em] \Rightarrow a.$

Hence, the required difference = $\bm{a}$.

Subtract the second expression from the first:

$-2b + 2c, b + 3c$

Answer

Solving,

$\Rightarrow (-2b + 2c) - (b + 3c) \\[1em] \Rightarrow -2b + 2c - b - 3c \\[1em] \Rightarrow -3b - c$

Hence, the required difference = $\bm{-3b - c}$.

Subtract the second expression from the first:

$5a + b, - 6b + 2a$

Answer

Solving,

$\Rightarrow (5a + b) - (-6b + 2a) \\[1em] \Rightarrow 5a + b + 6b - 2a \\[1em] \Rightarrow 3a + 7b$

Hence, the required difference = $\bm{3a + 7b}$.

Subtract the second expression from the first:

$a^3 - 1 + a, 3a - 2a^2$

Answer

Solving,

$\Rightarrow (a^3 - 1 + a) - (3a - 2a^2) \\[1em] \Rightarrow a^3 - 1 + a - 3a + 2a^2 \\[1em] \Rightarrow a^3 + 2a^2 - 2a - 1$

Hence, the required difference = $\bm{a^3 + 2a^2 - 2a - 1}$.

Subtract the second expression from the first:

$p + 2, 1$

Answer

Solving,

$\Rightarrow (p + 2) - 1 \\[1em] \Rightarrow p + 1$

Hence, the required difference = $\bm{p + 1}$.

Subtract the second expression from the first:

$x + 2y + z, - x - y - 3z$

Answer

Solving,

$\Rightarrow (x + 2y + z) - (-x - y - 3z) \\[1em] \Rightarrow x + 2y + z + x + y + 3z \\[1em] \Rightarrow 2x + 3y + 4z$

Hence, the required difference = $\bm{2x + 3y + 4z}$.

Subtract the second expression from the first:

$3a^2 - 8ab - 2b^2, 3a^2 - 4ab + 6b^2$

Answer

Solving,

$\Rightarrow (3a^2 - 8ab - 2b^2) - (3a^2 - 4ab + 6b^2) \\[1em] \Rightarrow 3a^2 - 8ab - 2b^2 - 3a^2 + 4ab - 6b^2 \\[1em] \Rightarrow -4ab - 8b^2$

Hence, the required difference = $\bm{-4ab - 8b^2}$.

Subtract the second expression from the first:

$4pq - 6p^2 - 2q^2, 9p^2$

Answer

Solving,

$\Rightarrow (4pq - 6p^2 - 2q^2) - 9p^2 \\[1em] \Rightarrow 4pq - 6p^2 - 2q^2 - 9p^2 \\[1em] \Rightarrow 4pq - 15p^2 - 2q^2$

Hence, the required difference = $\bm{4pq - 15p^2 - 2q^2}$.

Subtract the second expression from the first:

$10abc, 2a^2 + 2abc - 4b^2$

Answer

Solving,

$\Rightarrow 10abc - (2a^2 + 2abc - 4b^2) \\[1em] \Rightarrow 10abc - 2a^2 - 2abc + 4b^2 \\[1em] \Rightarrow -2a^2 + 8abc + 4b^2$

Hence, the required difference = $\bm{-2a^2 + 8abc + 4b^2}$.

Subtract the second expression from the first:

$a^2 + ab + c^2, a^2 - d^2$

Answer

Solving,

$\Rightarrow (a^2 + ab + c^2) - (a^2 - d^2) \\[1em] \Rightarrow a^2 + ab + c^2 - a^2 + d^2 \\[1em] \Rightarrow ab + c^2 + d^2$

Hence, the required difference = $\bm{ab + c^2 + d^2}$.

Subtract:

$4x$ from $8 - x$

Answer

Solving,

$\Rightarrow (8 - x) - 4x \\[1em] \Rightarrow 8 - x - 4x \\[1em] \Rightarrow 8 - 5x$

Hence, the required difference = $\bm{8 - 5x}$.

Subtract:

$-8c$ from $c + 3d$

Answer

Solving,

$\Rightarrow (c + 3d) - (-8c) \\[1em] \Rightarrow c + 3d + 8c \\[1em] \Rightarrow 9c + 3d$

Hence, the required difference = $\bm{9c + 3d}$.

Subtract:

$-5a - 2b$ from $b + 6c$

Answer

Solving,

$\Rightarrow (b + 6c) - (-5a - 2b) \\[1em] \Rightarrow b + 6c + 5a + 2b \\[1em] \Rightarrow 5a + 3b + 6c$

Hence, the required difference = $\bm{5a + 3b + 6c}$.

Subtract:

$4p + p^2$ from $3p^2 - 8p$

Answer

Solving,

$\Rightarrow (3p^2 - 8p) - (4p + p^2) \\[1em] \Rightarrow 3p^2 - 8p - 4p - p^2 \\[1em] \Rightarrow 2p^2 - 12p$

Hence, the required difference = $\bm{2p^2 - 12p}$.

Subtract:

$5a - 3b + 2c$ from $4a - b - 2c$

Answer

Solving,

$\Rightarrow (4a - b - 2c) - (5a - 3b + 2c) \\[1em] \Rightarrow 4a - b - 2c - 5a + 3b - 2c \\[1em] \Rightarrow -a + 2b - 4c$

Hence, the required difference = $\bm{-a + 2b - 4c}$.

Subtract:

$-xy + yz - zx$ from $xy - yz + xz$

Answer

Solving,

$\Rightarrow (xy - yz + xz) - (-xy + yz - zx) \\[1em] \Rightarrow xy - yz + xz + xy - yz + zx \\[1em] \Rightarrow 2xy - 2yz + 2xz \\[1em] \Rightarrow 2(xy - yz + xz)$

Hence, the required difference = $\bm{2(xy - yz + xz)}$.

Subtract:

$2x^2 - 7xy - y^2$ from $3x^2 - 5xy + 3y^2$

Answer

Solving,

$\Rightarrow (3x^2 - 5xy + 3y^2) - (2x^2 - 7xy - y^2) \\[1em] \Rightarrow 3x^2 - 5xy + 3y^2 - 2x^2 + 7xy + y^2 \\[1em] \Rightarrow x^2 + 2xy + 4y^2$

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

Subtract:

$a^2 - 3ab - 6b^2$ from $2b^2 - a^2 + 2ab$

Answer

Solving,

$\Rightarrow (2b^2 - a^2 + 2ab) - (a^2 - 3ab - 6b^2) \\[1em] \Rightarrow 2b^2 - a^2 + 2ab - a^2 + 3ab + 6b^2 \\[1em] \Rightarrow -2a^2 + 5ab + 8b^2$

Hence, the required difference = $\bm{-2a^2 + 5ab + 8b^2}$.

Subtract:

$4x^2 - 5x^2y + y^2$ from $- 3y^2 + 5xy^2 - 7x^2 - 9x^2y$

Answer

Solving,

$\Rightarrow (-3y^2 + 5xy^2 - 7x^2 - 9x^2y) - (4x^2 - 5x^2y + y^2) \\[1em] \Rightarrow -3y^2 + 5xy^2 - 7x^2 - 9x^2y - 4x^2 + 5x^2y - y^2 \\[1em] \Rightarrow (-7 - 4)x^2 + (-9 + 5)x^2y + 5xy^2 + (-3 - 1)y^2 \\[1em] \Rightarrow -11x^2 - 4x^2y + 5xy^2 - 4y^2$

Hence, the required difference = $\bm{-11x^2 - 4x^2y + 5xy^2 - 4y^2}$.

Subtract:

$6m^3 + 4m^2 + 7m - 3$ from $3m^3 + 4$

Answer

Solving,

$\Rightarrow (3m^3 + 4) - (6m^3 + 4m^2 + 7m - 3) \\[1em] \Rightarrow 3m^3 + 4 - 6m^3 - 4m^2 - 7m + 3 \\[1em] \Rightarrow -3m^3 - 4m^2 - 7m + 7$

Hence, the required difference = $\bm{-3m^3 - 4m^2 - 7m + 7}$.

Subtract $- 5a^2 - 3a + 1$ from the sum of $4a^2 + 3 - 8a$ and $9a - 7$.

Answer

Sum of $4a^2 + 3 - 8a$ and $9a - 7$,

$\Rightarrow (4a^2 + 3 - 8a) + (9a - 7) \\[1em] \Rightarrow 4a^2 + (-8 + 9)a + (3 - 7) \\[1em] \Rightarrow 4a^2 + a - 4$

Now, subtracting $-5a^2 - 3a + 1$ from this sum,

$\Rightarrow (4a^2 + a - 4) - (-5a^2 - 3a + 1) \\[1em] \Rightarrow 4a^2 + a - 4 + 5a^2 + 3a - 1 \\[1em] \Rightarrow 9a^2 + 4a - 5$

Hence, the required difference = $\bm{9a^2 + 4a - 5}$.

By how much does $8x^3 - 6x^2 + 9x - 10$ exceed $4x^3 + 2x^2 + 7x - 3$?

Answer

Solving,

$\Rightarrow (8x^3 - 6x^2 + 9x - 10) - (4x^3 + 2x^2 + 7x - 3) \\[1em] \Rightarrow 8x^3 - 6x^2 + 9x - 10 - 4x^3 - 2x^2 - 7x + 3 \\[1em] \Rightarrow (8 - 4)x^3 + (-6 - 2)x^2 + (9 - 7)x + (-10 + 3) \\[1em] \Rightarrow 4x^3 - 8x^2 + 2x - 7$

Hence, the required excess = $\bm{4x^3 - 8x^2 + 2x - 7}$.

What must be added to $2a^3 + 5a - a^2 - 6$ to get $a^2 - a - a^3 + 1$?

Answer

Solving,

$\Rightarrow (a^2 - a - a^3 + 1) - (2a^3 + 5a - a^2 - 6) \\[1em] \Rightarrow a^2 - a - a^3 + 1 - 2a^3 - 5a + a^2 + 6 \\[1em] \Rightarrow (-1 - 2)a^3 + (1 + 1)a^2 + (-1 - 5)a + (1 + 6) \\[1em] \Rightarrow -3a^3 + 2a^2 - 6a + 7$

Hence, the required expression = $\bm{-3a^3 + 2a^2 - 6a + 7}$.

What must be subtracted from $a^2 + b^2 + 2ab$ to get $- 4ab + 2b^2$?

Answer

Solving,

$\Rightarrow (a^2 + b^2 + 2ab) - (-4ab + 2b^2) \\[1em] \Rightarrow a^2 + b^2 + 2ab + 4ab - 2b^2 \\[1em] \Rightarrow a^2 + (1 - 2)b^2 + (2 + 4)ab \\[1em] \Rightarrow a^2 - b^2 + 6ab$

Hence, the required expression = $\bm{a^2 - b^2 + 6ab}$.

Find the excess of $4m^2 + 4n^2 + 4p^2$ over $m^2 + 3n^2 - 5p^2$.

Answer

Solving,

$\Rightarrow (4m^2 + 4n^2 + 4p^2) - (m^2 + 3n^2 - 5p^2) \\[1em] \Rightarrow 4m^2 + 4n^2 + 4p^2 - m^2 - 3n^2 + 5p^2 \\[1em] \Rightarrow (4 - 1)m^2 + (4 - 3)n^2 + (4 + 5)p^2 \\[1em] \Rightarrow 3m^2 + n^2 + 9p^2$

Hence, the required excess = $\bm{3m^2 + n^2 + 9p^2}$.

By how much is $3x^3 - 2x^2y + xy^2 - y^3$ less than $4x^3 - 3x^2y - 7xy^2 + 2y^3$?

Answer

Solving,

$\Rightarrow (4x^3 - 3x^2y - 7xy^2 + 2y^3) - (3x^3 - 2x^2y + xy^2 - y^3) \\[1em] \Rightarrow 4x^3 - 3x^2y - 7xy^2 + 2y^3 - 3x^3 + 2x^2y - xy^2 + y^3 \\[1em] \Rightarrow (4 - 3)x^3 + (-3 + 2)x^2y + (-7 - 1)xy^2 + (2 + 1)y^3 \\[1em] \Rightarrow x^3 - x^2y - 8xy^2 + 3y^3$

Hence, the required difference = $\bm{x^3 - x^2y - 8xy^2 + 3y^3}$.

Subtract the sum of $3a^2 - 2a + 5$ and $a^2 - 5a - 7$ from the sum of $5a^2 - 9a + 3$ and $2a - a^2 - 1$.

Answer

Sum of $5a^2 - 9a + 3$ and $2a - a^2 - 1$,

$\Rightarrow (5a^2 - 9a + 3) + (2a - a^2 - 1) \\[1em] \Rightarrow (5 - 1)a^2 + (-9 + 2)a + (3 - 1) \\[1em] \Rightarrow 4a^2 - 7a + 2$

Sum of $3a^2 - 2a + 5$ and $a^2 - 5a - 7$,

$\Rightarrow (3a^2 - 2a + 5) + (a^2 - 5a - 7) \\[1em] \Rightarrow (3 + 1)a^2 + (-2 - 5)a + (5 - 7) \\[1em] \Rightarrow 4a^2 - 7a - 2$

Now, subtracting the second sum from the first sum,

$\Rightarrow (4a^2 - 7a + 2) - (4a^2 - 7a - 2) \\[1em] \Rightarrow 4a^2 - 7a + 2 - 4a^2 + 7a + 2 \\[1em] \Rightarrow 4$

Hence, the required difference = $\bm{4}$.

The perimeter of a rectangle is $28x^3 + 16x^2 + 8x + 4$. One of its sides is $8x^2 + 4x$. Find the other side.

Answer

Let the length and the breadth of the rectangle be $l$ and $b$.

Perimeter = $2(l + b)$

$\Rightarrow l + b = \dfrac{\text{Perimeter}}{2} \\[1em] \Rightarrow l + b = \dfrac{28x^3 + 16x^2 + 8x + 4}{2} \\[1em] \Rightarrow l + b = 14x^3 + 8x^2 + 4x + 2$