Algebraic Expressions and Identities — Exercise 10.2

Find the product of:

$4x^3$ and $-3xy$

Answer

Product of two or more monomials = (product of their numerical coefficients) × (product of their literal coefficients)

$\Rightarrow 4x^3 \times (-3xy) \\[1em] \Rightarrow (4 \times (-3)) \times (x^3 \times x) \times y \\[1em] \Rightarrow -12x^{(3+1)}y \\[1em] \Rightarrow -12x^4y$

Hence, the required product = $\bm{-12x^4y}$.

Find the product of:

$2xyz$ and 0

Answer

Product of two or more monomials = (product of their numerical coefficients) × (product of their literal coefficients)

$\Rightarrow 2xyz \times 0 \\[1em] \Rightarrow (2 \times 0) \times xyz \\[1em] \Rightarrow 0$

Hence, the required product = 0.

Find the product of:

$-\dfrac{2}{3}p^2q, \dfrac{3}{4}pq^2$ and $5pqr$

Answer

Product of two or more monomials = (product of their numerical coefficients) × (product of their literal coefficients)

$\Rightarrow -\dfrac{2}{3}p^2q \times \dfrac{3}{4}pq^2 \times 5pqr \\[1em] \Rightarrow \Big(-\dfrac{2}{3} \times \dfrac{3}{4} \times 5\Big) \times p^{2+1+1} \times q^{1+2+1} \times r \\[1em] \Rightarrow -\dfrac{2 \times 3 \times 5}{3 \times 4} \times p^4q^4r \\[1em] \Rightarrow -\dfrac{30}{12}p^4q^4r \\[1em] \Rightarrow -\dfrac{5}{2}p^4q^4r$

Hence, the required product = $\bm{-\dfrac{5}{2}p^4q^4r}$.

Find the product of:

$-\dfrac{1}{2}x^2, -\dfrac{3}{5}xy, \dfrac{2}{3}yz$ and $\dfrac{5}{7}xyz$

Answer

Product of two or more monomials = (product of their numerical coefficients) × (product of their literal coefficients)

$\Rightarrow -\dfrac{1}{2}x^2 \times \Big(-\dfrac{3}{5}xy\Big) \times \dfrac{2}{3}yz \times \dfrac{5}{7}xyz \\[1em] \Rightarrow \Big(-\dfrac{1}{2} \times -\dfrac{3}{5} \times \dfrac{2}{3} \times \dfrac{5}{7}\Big) \times x^{2+1+1} \times y^{1+1+1} \times z^{1+1} \\[1em] \Rightarrow \dfrac{1 \times 3 \times 2 \times 5}{2 \times 5 \times 3 \times 7} \times x^4y^3z^2 \\[1em] \Rightarrow \dfrac{30}{210}x^4y^3z^2 \\[1em] \Rightarrow \dfrac{1}{7}x^4y^3z^2$

Hence, the required product = $\bm{\dfrac{1}{7}x^4y^3z^2}$.

Find the area of rectangle with the following pair of monomials as its length and breadth respectively:

$(p^2q, pq^2)$

Answer

Area of rectangle = length × breadth

$\Rightarrow p^2q \times pq^2 \\[1em] \Rightarrow (p^2 \times p) \times (q \times q^2) \\[1em] \Rightarrow p^{(2+1)}q^{(1+2)} \\[1em] \Rightarrow p^3q^3$

Hence, the area of the rectangle = $\bm{p^3q^3}$.

Find the area of rectangle with the following pair of monomials as its length and breadth respectively:

$(5xy, 7xy^2)$

Answer

Area of rectangle = length × breadth

$\Rightarrow 5xy \times 7xy^2 \\[1em] \Rightarrow (5 \times 7) \times (x \times x) \times (y \times y^2) \\[1em] \Rightarrow 35x^{(1+1)}y^{(1+2)} \\[1em] \Rightarrow 35x^2y^3$

Hence, the area of the rectangle = $\bm{35x^2y^3}$.

Find the volume of a rectangular box with the following length, breadth and height respectively:

$5ab, 3a^2b, 7a^4b^2$

Answer

Volume of a rectangular box = length × breadth × height

$\Rightarrow 5ab \times 3a^2b \times 7a^4b^2 \\[1em] \Rightarrow (5 \times 3 \times 7) \times (a \times a^2 \times a^4) \times (b \times b \times b^2) \\[1em] \Rightarrow 105a^{(1+2+4)}b^{(1+1+2)} \\[1em] \Rightarrow 105a^7b^4$

Hence, the volume of the rectangular box = $\bm{105a^7b^4}$.

Find the volume of a rectangular box with the following length, breadth and height respectively:

$2pq, 4q^2, 8rp$

Answer

Volume of a rectangular box = length × breadth × height

$\Rightarrow 2pq \times 4q^2 \times 8rp \\[1em] \Rightarrow (2 \times 4 \times 8) \times (p \times p) \times (q \times q^2) \times r \\[1em] \Rightarrow 64p^{(1+1)}q^{(1+2)}r \\[1em] \Rightarrow 64p^2q^3r$

Hence, the volume of the rectangular box = $\bm{64p^2q^3r}$.

Multiply:

$(3x - 5y + 7z)$ by $-3xyz$

Answer

To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

$\Rightarrow (3x - 5y + 7z) \times (-3xyz) \\[1em] \Rightarrow 3x \times (-3xyz) - 5y \times (-3xyz) + 7z \times (-3xyz) \\[1em] \Rightarrow -9x^{(1+1)}yz + 15xy^{(1+1)}z - 21xyz^{(1+1)} \\[1em] \Rightarrow -9x^2yz + 15xy^2z - 21xyz^2$

Hence, the required product = $\bm{-9x^2yz + 15xy^2z - 21xyz^2}$.

Multiply:

$(2p^2 - 3pq + 5q^2 + 5)$ by $-2pq$

Answer

To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

$\Rightarrow (2p^2 - 3pq + 5q^2 + 5) \times (-2pq) \\[1em] \Rightarrow 2p^2 \times (-2pq) - 3pq \times (-2pq) + 5q^2 \times (-2pq) + 5 \times (-2pq) \\[1em] \Rightarrow -4p^{(2+1)}q + 6p^{(1+1)}q^{(1+1)} - 10pq^{(2+1)} - 10pq \\[1em] \Rightarrow -4p^3q + 6p^2q^2 - 10pq^3 - 10pq$

Hence, the required product = $\bm{-4p^3q + 6p^2q^2 - 10pq^3 - 10pq}$.

Multiply:

$\left(\dfrac{2}{3}a^2b - \dfrac{4}{5}ab^2 + \dfrac{2}{7}ab + 3\right)$ by $35ab$

Answer

To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

$\Rightarrow \Big(\dfrac{2}{3}a^2b - \dfrac{4}{5}ab^2 + \dfrac{2}{7}ab + 3\Big) \times 35ab \\[1em] \Rightarrow \dfrac{2}{3}a^2b \times 35ab - \dfrac{4}{5}ab^2 \times 35ab + \dfrac{2}{7}ab \times 35ab + 3 \times 35ab \\[1em] \Rightarrow \dfrac{2 \times 35}{3}a^3b^2 - \dfrac{4 \times 35}{5}a^2b^3 + \dfrac{2 \times 35}{7}a^2b^2 + 105ab \\[1em] \Rightarrow \dfrac{70}{3}a^3b^2 - 28a^2b^3 + 10a^2b^2 + 105ab$

Hence, the required product = $\bm{\dfrac{70}{3}a^3b^2 - 28a^2b^3 + 10a^2b^2 + 105ab}$.

Multiply:

$(4x^2 - 10xy + 7y^2 - 8x + 4y + 3)$ by $3xy$

Answer

To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

$\Rightarrow (4x^2 - 10xy + 7y^2 - 8x + 4y + 3) \times 3xy \\[1em] \Rightarrow 4x^2 \times 3xy - 10xy \times 3xy + 7y^2 \times 3xy - 8x \times 3xy + 4y \times 3xy + 3 \times 3xy \\[1em] \Rightarrow 12x^3y - 30x^2y^2 + 21xy^3 - 24x^2y + 12xy^2 + 9xy$

Hence, the required product =

$\bm{12x^3y - 30x^2y^2 + 21xy^3 - 24x^2y + 12xy^2 + 9xy}$.

Simplify the following expression and evaluate it as directed:

$x^2(3 - 2x + x^2)$ for $x = 1$; $x = -1$; $x = \dfrac{2}{3}$ and $x = -\dfrac{1}{2}$

Answer

First we simplify the expression by multiplying each term of the polynomial by the monomial.

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

For $x = 1$, the value of $x^4 - 2x^3 + 3x^2$ is

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

For $x = -1$, the value of $x^4 - 2x^3 + 3x^2$ is

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

For $x = \dfrac{2}{3}$, the value of $x^4 - 2x^3 + 3x^2$ is

$\Rightarrow \Big(\dfrac{2}{3}\Big)^4 - 2\Big(\dfrac{2}{3}\Big)^3 + 3\Big(\dfrac{2}{3}\Big)^2 \\[1em] \Rightarrow \dfrac{16}{81} - 2 \times \dfrac{8}{27} + 3 \times \dfrac{4}{9} \\[1em] \Rightarrow \dfrac{16}{81} - \dfrac{16}{27} + \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{16 - 48 + 108}{81} \\[1em] \Rightarrow \dfrac{76}{81}$

For $x = -\dfrac{1}{2}$, the value of $x^4 - 2x^3 + 3x^2$ is

$\Rightarrow \Big(-\dfrac{1}{2}\Big)^4 - 2\Big(-\dfrac{1}{2}\Big)^3 + 3\Big(-\dfrac{1}{2}\Big)^2 \\[1em] \Rightarrow \dfrac{1}{16} + 2 \times \dfrac{1}{8} + 3 \times \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{1}{16} + \dfrac{1}{4} + \dfrac{3}{4} \\[1em] \Rightarrow \dfrac{1 + 4 + 12}{16} \\[1em] \Rightarrow \dfrac{17}{16}$

Hence, $\bm{x^2(3 - 2x + x^2) = x^4 - 2x^3 + 3x^2}$ and its values are 2, 6, $\bm{\dfrac{76}{81}}$ and $\bm{\dfrac{17}{16}}$ respectively.

Simplify the following expression and evaluate it as directed:

$5xy(3x + 4y - 7) - 3y(xy - x^2 + 9) - 8$ for $x = 2$, $y = -1$

Answer

First we simplify the expression by multiplying each polynomial by the monomial outside it.

$\Rightarrow 5xy(3x + 4y - 7) - 3y(xy - x^2 + 9) - 8 \\[1em] \Rightarrow 15x^2y + 20xy^2 - 35xy - 3xy^2 + 3x^2y - 27y - 8 \\[1em] \Rightarrow (15x^2y + 3x^2y) + (20xy^2 - 3xy^2) - 35xy - 27y - 8 \\[1em] \Rightarrow 18x^2y + 17xy^2 - 35xy - 27y - 8$

For $x = 2$, $y = -1$, the value of $18x^2y + 17xy^2 - 35xy - 27y - 8$ is

$\Rightarrow 18(2)^2(-1) + 17(2)(-1)^2 - 35(2)(-1) - 27(-1) - 8 \\[1em] \Rightarrow 18 \times 4 \times (-1) + 17 \times 2 \times 1 + 70 + 27 - 8 \\[1em] \Rightarrow -72 + 34 + 70 + 27 - 8 \\[1em] \Rightarrow 51$

Hence, the simplified expression = $\bm{18x^2y + 17xy^2 - 35xy - 27y - 8}$ and its value = 51.

Add the following:

$4p(2 - p^2)$ and $8p^3 - 3p$

Answer

First we simplify the given expressions and then collect the like terms.

$\Rightarrow 4p(2 - p^2) + 8p^3 - 3p \\[1em] \Rightarrow 8p - 4p^3 + 8p^3 - 3p \\[1em] \Rightarrow (-4p^3 + 8p^3) + (8p - 3p) \\[1em] \Rightarrow 4p^3 + 5p$

Hence, the required sum = $\bm{4p^3 + 5p}$.

Add the following:

$7xy(8x + 2y - 3)$ and $4xy^2(3y - 7x + 8)$

Answer

First we simplify the given expressions and then collect the like terms.

$\Rightarrow 7xy(8x + 2y - 3) + 4xy^2(3y - 7x + 8) \\[1em] \Rightarrow 56x^2y + 14xy^2 - 21xy + 12xy^3 - 28x^2y^2 + 32xy^2 \\[1em] \Rightarrow 12xy^3 - 28x^2y^2 + 56x^2y + (14xy^2 + 32xy^2) - 21xy \\[1em] \Rightarrow 12xy^3 - 28x^2y^2 + 56x^2y + 46xy^2 - 21xy$

Hence, the required sum = $\bm{12xy^3 - 28x^2y^2 + 56x^2y + 46xy^2 - 21xy}$.

Subtract:

$6x(x - y + z) - 3y(x + y - z)$ from $2z(-x + y + z)$

Answer

First we simplify both the expressions and then subtract.

Simplifying first expression:

$\Rightarrow 6x(x - y + z) - 3y(x + y - z) \\[1em] \Rightarrow 6x^2 - 6xy + 6zx - 3xy - 3y^2 + 3yz \\[1em] \Rightarrow 6x^2 - 9xy - 3y^2 + 3yz + 6zx$

Simplifying second expression:

$\Rightarrow 2z(-x + y + z) \\[1em] \Rightarrow -2zx + 2yz + 2z^2$

Subtracting first expression from second expression:

$\Rightarrow (-2zx + 2yz + 2z^2) - (6x^2 - 9xy - 3y^2 + 3yz + 6zx) \\[1em] \Rightarrow -2zx + 2yz + 2z^2 - 6x^2 + 9xy + 3y^2 - 3yz - 6zx \\[1em] \Rightarrow -6x^2 + 3y^2 + 2z^2 + 9xy + (2yz - 3yz) + (-2zx - 6zx) \\[1em] \Rightarrow -6x^2 + 3y^2 + 2z^2 + 9xy - yz - 8zx$

Hence, the required difference = $\bm{-6x^2 + 3y^2 + 2z^2 + 9xy - yz - 8zx}$.

Subtract:

$7xy(x^2 - 2xy + 3y^2) - 8x(x^2y - 4xy + 7xy^2)$ from $3y(4x^2y - 5xy + 8xy^2)$

Answer

First we simplify both the expressions and then subtract.

Simplifying first expression:

$\Rightarrow 7xy(x^2 - 2xy + 3y^2) - 8x(x^2y - 4xy + 7xy^2) \\[1em] \Rightarrow 7x^3y - 14x^2y^2 + 21xy^3 - 8x^3y + 32x^2y - 56x^2y^2 \\[1em] \Rightarrow (7x^3y - 8x^3y) + (-14x^2y^2 - 56x^2y^2) + 21xy^3 + 32x^2y \\[1em] \Rightarrow -x^3y - 70x^2y^2 + 21xy^3 + 32x^2y$

Simplifying second expression:

$\Rightarrow 3y(4x^2y - 5xy + 8xy^2) \\[1em] \Rightarrow 12x^2y^2 - 15xy^2 + 24xy^3$

Subtracting first expression from second expression:

$\Rightarrow (12x^2y^2 - 15xy^2 + 24xy^3) - (-x^3y - 70x^2y^2 + 21xy^3 + 32x^2y) \\[1em] \Rightarrow 12x^2y^2 - 15xy^2 + 24xy^3 + x^3y + 70x^2y^2 - 21xy^3 - 32x^2y \\[1em] \Rightarrow (12x^2y^2 + 70x^2y^2) + (24xy^3 - 21xy^3) + x^3y - 15xy^2 - 32x^2y \\[1em] \Rightarrow 82x^2y^2 + 3xy^3 + x^3y - 15xy^2 - 32x^2y$

Hence, the required difference = $\bm{82x^2y^2 + 3xy^3 + x^3y - 15xy^2 - 32x^2y}$.