Fundamental Concepts of Algebra — Exercise 12(D)

Fill in the blanks:

(i) $6 \times 3$ = ............... and $6x \times 3x$ = ...............

(ii) $6 \times 3$ = ............... and $6x^2 \times 3x^3$ = ...............

(iii) $5 \times 4$ = ............... and $5x \times 4y$ = ...............

(iv) $4 \times 7$ = ............... and $4ax \times 7x$ = ...............

(v) $6 \times 2$ = ............... and $6xy \times 2xy$ = ...............

(vi) $12 \times 4$ = ............... and $12ax^2 \times 4ax$ = ...............

(vii) $1 \times 8$ = ............... and $a^2xy^2 \times 8a^3x^2y$ = ...............

(viii) $15 \times 3$ = ............... and $15x \times 3x^5y^2$ = ...............

Answer

The product of the given monomials = (product of their numeral coefficients) × (product of their literals), and in multiplication the powers of like factors are added.

(i) $6 \times 3 =$ 18 and $6x \times 3x = (6 \times 3)(x \times x) = \bm{18x^2}$.

(ii) $6 \times 3 =$ 18 and $6x^2 \times 3x^3 = (6 \times 3)(x^{2+3}) = \bm{18x^5}$.

(iii) $5 \times 4 =$ 20 and $5x \times 4y = (5 \times 4)(x \times y) = \bm{20xy}$.

(iv) $4 \times 7 =$ 28 and $4ax \times 7x = (4 \times 7)(a \times x^{1+1}) = \bm{28ax^2}$.

(v) $6 \times 2 =$ 12 and $6xy \times 2xy = (6 \times 2)(x^{1+1} \times y^{1+1}) = \bm{12x^2y^2}$.

(vi) $12 \times 4 =$ 48 and $12ax^2 \times 4ax = (12 \times 4)(a^{1+1} \times x^{2+1}) = \bm{48a^2x^3}$.

(vii) $1 \times 8 =$ 8 and $a^2xy^2 \times 8a^3x^2y = (1 \times 8)(a^{2+3} \times x^{1+2} \times y^{2+1}) = \bm{8a^5x^3y^3}$.

(viii) $15 \times 3 =$ 45 and $15x \times 3x^5y^2 = (15 \times 3)(x^{1+5} \times y^2) = \bm{45x^6y^2}$.

Fill in the blanks:

(i) $4x \times 6x \times 2$ = ...............

(ii) $3ab \times 6ax$ = ...............

(iii) $x \times 2x^2 \times 3x^3$ = ...............

(iv) $5 \times 5a^3$ = ...............

(v) $6 \times 6x^2 \times 6x^2y^2$ = ...............

(vi) $−8x \times −3x$ = ...............

(vii) $−5 \times −3x \times 5x^2$ = ...............

(viii) $8 \times −4xy^2 \times 3x^3y^2$ = ...............

(ix) $−4x \times 5xy \times 3z$ = ...............

(x) $5x \times 2x^2y \times −7y^3 \times 2x^3y^2$ = ...............

Answer

The product of the given monomials = (product of their numeral coefficients) × (product of their literals), and in multiplication the powers of like factors are added.

(i) $4x \times 6x \times 2 = (4 \times 6 \times 2)(x^{1+1}) = \bm{48x^2}$.

(ii) $3ab \times 6ax = (3 \times 6)(a^{1+1} \times b \times x) = \bm{18a^2bx}$.

(iii) $x \times 2x^2 \times 3x^3 = (1 \times 2 \times 3)(x^{1+2+3}) = \bm{6x^6}$.

(iv) $5 \times 5a^3 = (5 \times 5)a^3 = \bm{25a^3}$.

(v) $6 \times 6x^2 \times 6x^2y^2 = (6 \times 6 \times 6)(x^{2+2} \times y^2) = \bm{216x^4y^2}$.

(vi) $-8x \times -3x = (-8 \times -3)(x^{1+1}) = \bm{24x^2}$.

(vii) $-5 \times -3x \times 5x^2 = (-5 \times -3 \times 5)(x^{1+2}) = \bm{75x^3}$.

(viii) $8 \times -4xy^2 \times 3x^3y^2 = (8 \times -4 \times 3)(x^{1+3} \times y^{2+2}) = \bm{-96x^4y^4}$.

(ix) $-4x \times 5xy \times 3z = (-4 \times 5 \times 3)(x^{1+1} \times y \times z) = \bm{-60x^2yz}$.

(x) $5x \times 2x^2y \times -7y^3 \times 2x^3y^2 = (5 \times 2 \times -7 \times 2)(x^{1+2+3} \times y^{1+3+2}) = \bm{-140x^6y^6}$.

Find the value of:

$3x^3 \times 5x^4$

Answer

Solving,

$\Rightarrow 3x^3 \times 5x^4 \\[1em] \Rightarrow (3 \times 5)(x^{3+4}) \\[1em] \Rightarrow 15x^7$

Hence, $\bm{3x^3 \times 5x^4 = 15x^7}$.

Find the value of:

$5a^2 \times 7a^7$

Answer

Solving,

$\Rightarrow 5a^2 \times 7a^7 \\[1em] \Rightarrow (5 \times 7)(a^{2+7}) \\[1em] \Rightarrow 35a^9$

Hence, $\bm{5a^2 \times 7a^7 = 35a^9}$.

Find the value of:

$3abc \times 6ac^3$

Answer

Solving,

$\Rightarrow 3abc \times 6ac^3 \\[1em] \Rightarrow (3 \times 6)(a^{1+1} \times b \times c^{1+3}) \\[1em] \Rightarrow 18a^2bc^4$

Hence, $\bm{3abc \times 6ac^3 = 18a^2bc^4}$.

Find the value of:

$a^2b^2 \times 5a^3b^4$

Answer

Solving,

$\Rightarrow a^2b^2 \times 5a^3b^4 \\[1em] \Rightarrow (1 \times 5)(a^{2+3} \times b^{2+4}) \\[1em] \Rightarrow 5a^5b^6$

Hence, $\bm{a^2b^2 \times 5a^3b^4 = 5a^5b^6}$.

Find the value of:

$2x^2y^3 \times 5x^3y^4$

Answer

Solving,

$\Rightarrow 2x^2y^3 \times 5x^3y^4 \\[1em] \Rightarrow (2 \times 5)(x^{2+3} \times y^{3+4}) \\[1em] \Rightarrow 10x^5y^7$

Hence, $\bm{2x^2y^3 \times 5x^3y^4 = 10x^5y^7}$.

Find the value of:

$abc \times bcd$

Answer

Solving,

$\Rightarrow abc \times bcd \\[1em] \Rightarrow a \times b^{1+1} \times c^{1+1} \times d \\[1em] \Rightarrow ab^2c^2d$

Hence, $\bm{abc \times bcd = ab^2c^2d}$.

Multiply:

$a + b$ by $ab$

Answer

Solving,

$\begin{array}{rrrr} &a &+& b \\ \times & &&ab \\ \hline &a^2b &+& ab^2 \end{array}$

Hence, $\bm{(a + b) \times ab = a^2b + ab^2}$.

Multiply:

$3ab - 4b$ by $3ab$

Answer

Solving,

$\begin{array}{rrrr} &3ab &-& 4b \\ \times & &&3ab \\ \hline &9a^2b^2 &-& 12ab^2 \end{array}$

Hence, $\bm{(3ab - 4b) \times 3ab = 9a^2b^2 - 12ab^2}$.

Multiply:

$2xy - 5by$ by $4bx$

Answer

Solving,

$\begin{array}{rrrr} &2xy &-& 5by \\ \times & &&4bx \\ \hline &8bx^2y &-& 20b^2xy \end{array}$

Hence, $\bm{(2xy - 5by) \times 4bx = 8bx^2y - 20b^2xy}$.

Multiply:

$4x + 2y$ by $3xy$

Answer

Solving,

$\begin{array}{rrrr} &4x &+& 2y \\ \times & &&3xy \\ \hline &12x^2y &+& 6xy^2 \end{array}$

Hence, $\bm{(4x + 2y) \times 3xy = 12x^2y + 6xy^2}$.

Multiply:

$x^2 - x$ by $2x$

Answer

Solving,

$\begin{array}{rrrr} &x^2 &-& x \\ \times & &&2x \\ \hline &2x^3 &-& 2x^2 \end{array}$

Hence, $\bm{(x^2 - x) \times 2x = 2x^3 - 2x^2}$.

Multiply:

$1 + 4x$ by $x$

Answer

Solving,

$\begin{array}{rrrr} &1 &+& 4x \\ \times & &&x \\ \hline &x &+& 4x^2 \end{array}$

Hence, $\bm{(1 + 4x) \times x = x + 4x^2}$.

Multiply:

$9xy^2 + 3x^2y$ by $5xy$

Answer

Solving,

$\begin{array}{rrrr} &9xy^2 &+& 3x^2y \\ \times & &&5xy \\ \hline &45x^2y^3 &+& 15x^3y^2 \end{array}$

Hence, $\bm{(9xy^2 + 3x^2y) \times 5xy = 45x^2y^3 + 15x^3y^2}$.

Multiply:

$6x - 5y$ by $3axy$

Answer

Solving,

$\begin{array}{rrrr} &6x &-& 5y \\ \times & &&3axy \\ \hline &18ax^2y &-& 15axy^2 \end{array}$

Hence, $\bm{(6x - 5y) \times 3axy = 18ax^2y - 15axy^2}$.

Multiply:

$-x + y - z$ and $-2x$

Answer

Solving,

$\begin{array}{rrrrrr} &-x &+& y &-& z \\ \times & &&&&-2x \\ \hline &2x^2 &-& 2xy &+& 2xz \end{array}$

Hence, $\bm{(-x + y - z) \times (-2x) = 2x^2 - 2xy + 2xz}$.

Multiply:

$xy - yz$ and $x^2yz^2$

Answer

Solving,

$\begin{array}{rrrr} &xy &-& yz \\ \times & &&x^2yz^2 \\ \hline &x^3y^2z^2 &-& x^2y^2z^3 \end{array}$

Hence, $\bm{(xy - yz) \times x^2yz^2 = x^3y^2z^2 - x^2y^2z^3}$.

Multiply:

$2xyz + 3xy$ and $-2y^2z$

Answer

Solving,

$\begin{array}{rrrr} &2xyz &+& 3xy \\ \times & &&-2y^2z \\ \hline &-4xy^3z^2 &-& 6xy^3z \end{array}$

Hence, $\bm{(2xyz + 3xy) \times (-2y^2z) = -4xy^3z^2 - 6xy^3z}$.

Multiply:

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

Answer

Solving,

$\begin{array}{rrrr} &-3xy^2 &+& 4x^2y \\ \times & &&-xy \\ \hline &3x^2y^3 &-& 4x^3y^2 \end{array}$

Hence, $\bm{(-3xy^2 + 4x^2y) \times (-xy) = 3x^2y^3 - 4x^3y^2}$.

Multiply:

$4xy$ and $-x^2y - 3x^2y^2$

Answer

Solving,

$\begin{array}{rrrr} &-x^2y &-& 3x^2y^2 \\ \times & &&4xy \\ \hline &-4x^3y^2 &-& 12x^3y^3 \end{array}$

Hence, $\bm{4xy \times (-x^2y - 3x^2y^2) = -4x^3y^2 - 12x^3y^3}$.

Multiply:

$3a + 4b - 5c$ and $3a$

Answer

Solving,

$\begin{array}{rrrrrr} &3a &+& 4b &-& 5c \\ \times & &&&&3a \\ \hline &9a^2 &+& 12ab &-& 15ac \end{array}$

Hence, $\bm{(3a + 4b - 5c) \times 3a = 9a^2 + 12ab - 15ac}$.

Multiply:

$-5xy$ and $-xy^2 - 6x^2y$

Answer

Solving,

$\begin{array}{rrrr} &-xy^2 &-& 6x^2y \\ \times & &&-5xy \\ \hline &5x^2y^3 &+& 30x^3y^2 \end{array}$

Hence, $\bm{-5xy \times (-xy^2 - 6x^2y) = 5x^2y^3 + 30x^3y^2}$.

Multiply:

$x + 2$ and $x + 10$

Answer

Solving,

$\Rightarrow (x + 2)(x + 10) \\[1em] \Rightarrow x^2 + 10x + 2x + 20 \\[1em] \Rightarrow x^2 + 12x + 20.$

Hence, $\bm{(x + 2)(x + 10) = x^2 + 12x + 20}$.

Multiply:

$x + 5$ and $x - 3$

Answer

Solving,

$\Rightarrow (x + 5)(x - 3) \\[1em] \Rightarrow x^2 - 3x + 5x - 15 \\[1em] \Rightarrow x^2 + 2x - 15.$

Hence, $\bm{(x + 5)(x - 3) = x^2 + 2x - 15}$.

Multiply:

$x - 5$ and $x + 3$

Answer

Solving,

$\Rightarrow (x - 5)(x + 3) \\[1em] \Rightarrow x^2 + 3x - 5x - 15 \\[1em] \Rightarrow x^2 - 2x - 15.$

Hence, $\bm{(x - 5)(x + 3) = x^2 - 2x - 15}$.

Multiply:

$x - 5$ and $x - 3$

Answer

Solving,

$\Rightarrow (x - 5)(x - 3) \\[1em] \Rightarrow x^2 - 3x - 5x + 15 \\[1em] \Rightarrow x^2 - 8x + 15.$

Hence, $\bm{(x - 5)(x - 3) = x^2 - 8x + 15}$.

Multiply:

$2x + y$ and $x + 3y$

Answer

Solving,

$\Rightarrow (2x + y)(x + 3y) \\[1em] \Rightarrow 2x^2 + 6xy + xy + 3y^2 \\[1em] \Rightarrow 2x^2 + 7xy + 3y^2.$

Hence, $\bm{(2x + y)(x + 3y) = 2x^2 + 7xy + 3y^2}$.

Multiply:

$3x - 5y$ and $x + 6y$

Answer

Solving,

$\Rightarrow (3x - 5y)(x + 6y) \\[1em] \Rightarrow 3x^2 + 18xy - 5xy - 30y^2 \\[1em] \Rightarrow 3x^2 + 13xy - 30y^2.$

Hence, $\bm{(3x - 5y)(x + 6y) = 3x^2 + 13xy - 30y^2}$.

Multiply:

$x + 9y$ and $x - 5y$

Answer

Solving,

$\Rightarrow (x + 9y)(x - 5y) \\[1em] \Rightarrow x^2 - 5xy + 9xy - 45y^2 \\[1em] \Rightarrow x^2 + 4xy - 45y^2.$

Hence, $\bm{(x + 9y)(x - 5y) = x^2 + 4xy - 45y^2}$.

Multiply:

$2x + 5y$ and $2x + 5y$

Answer

Solving,

$\Rightarrow (2x + 5y)(2x + 5y) \\[1em] \Rightarrow 4x^2 + 10xy + 10xy + 25y^2 \\[1em] \Rightarrow 4x^2 + 20xy + 25y^2.$

Hence, $\bm{(2x + 5y)(2x + 5y) = 4x^2 + 20xy + 25y^2}$.

Multiply:

$3abc$ and $-5a^2b^2c$

Answer

Solving,

$\begin{array}{rr} &3abc \\ \times &-5a^2b^2c \\ \hline &-15a^3b^3c^2 \end{array}$

Hence, $\bm{3abc \times (-5a^2b^2c) = -15a^3b^3c^2}$.

Multiply:

$x - y + z$ and $-2x$

Answer

Solving,

$\begin{array}{rrrrrr} &x &-& y &+& z \\ \times & &&&&-2x \\ \hline &-2x^2 &+& 2xy &-& 2xz \end{array}$

Hence, $\bm{(x - y + z) \times (-2x) = -2x^2 + 2xy - 2xz}$.

Multiply:

$2x - 3y - 5z$ and $-2y$

Answer

Solving,

$\begin{array}{rrrrrr} &2x &-& 3y &-& 5z \\ \times & &&&&-2y \\ \hline &-4xy &+& 6y^2 &+& 10yz \end{array}$

Hence, $\bm{(2x - 3y - 5z) \times (-2y) = -4xy + 6y^2 + 10yz}$.

Multiply:

$-8xyz + 10x^2yz^3$ and $xyz$

Answer

Solving,

$\begin{array}{rrrr} &-8xyz &+& 10x^2yz^3 \\ \times & &&xyz \\ \hline &-8x^2y^2z^2 &+& 10x^3y^2z^4 \end{array}$

Hence, $\bm{(-8xyz + 10x^2yz^3) \times xyz = -8x^2y^2z^2 + 10x^3y^2z^4}$.

Multiply:

$xyz$ and $-13xy^2z + 15x^2yz - 6xyz^2$

Answer

Solving,

$\begin{array}{rrrrrr} &-13xy^2z &+& 15x^2yz &-& 6xyz^2 \\ \times & &&&&xyz \\ \hline &-13x^2y^3z^2 &+& 15x^3y^2z^2 &-& 6x^2y^2z^3 \end{array}$

Hence, $\bm{xyz \times (-13xy^2z + 15x^2yz - 6xyz^2) = -13x^2y^3z^2 + 15x^3y^2z^2 - 6x^2y^2z^3}$.

Multiply:

$4abc - 5a^2bc - 6ab^2c$ and $-2abc^2$

Answer

Solving,

$\begin{array}{rrrrrr} &4abc &-& 5a^2bc &-& 6ab^2c \\ \times & &&&&-2abc^2 \\ \hline &-8a^2b^2c^3 &+& 10a^3b^2c^3 &+& 12a^2b^3c^3 \end{array}$

Hence, $\bm{(4abc - 5a^2bc - 6ab^2c) \times (-2abc^2) = -8a^2b^2c^3 + 10a^3b^2c^3 + 12a^2b^3c^3}$.

Find the product of:

$xy - ab$ and $xy + ab$

Answer

Solving,

$\Rightarrow (xy - ab)(xy + ab) \\[1em] \Rightarrow xy \times xy + xy \times ab - ab \times xy - ab \times ab \\[1em] \Rightarrow x^2y^2 + abxy - abxy - a^2b^2 \\[1em] \Rightarrow x^2y^2 - a^2b^2.$

Hence, $\bm{(xy - ab)(xy + ab) = x^2y^2 - a^2b^2}$.

Find the product of:

$2abc - 3xy$ and $2abc + 3xy$

Answer

Solving,

$\Rightarrow (2abc - 3xy)(2abc + 3xy) \\[1em] \Rightarrow 2abc \times 2abc + 2abc \times 3xy - 3xy \times 2abc - 3xy \times 3xy \\[1em] \Rightarrow 4a^2b^2c^2 + 6abcxy - 6abcxy - 9x^2y^2 \\[1em] \Rightarrow 4a^2b^2c^2 - 9x^2y^2.$

Hence, $\bm{(2abc - 3xy)(2abc + 3xy) = 4a^2b^2c^2 - 9x^2y^2}$.

Find the product of:

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

Answer

Solving,

$\Rightarrow (a + b - c)(2a - 3b) \\[1em] \Rightarrow a(2a - 3b) + b(2a - 3b) - c(2a - 3b) \\[1em] \Rightarrow 2a^2 - 3ab + 2ab - 3b^2 - 2ac + 3bc \\[1em] \Rightarrow 2a^2 - ab - 2ac - 3b^2 + 3bc.$

Hence, $\bm{(a + b - c)(2a - 3b) = 2a^2 - ab - 2ac - 3b^2 + 3bc}$.

Find the product of:

$5x - 6y - 7z$ and $2x + 3y$

Answer

Solving,

$\Rightarrow (5x - 6y - 7z)(2x + 3y) \\[1em] \Rightarrow 5x(2x + 3y) - 6y(2x + 3y) - 7z(2x + 3y) \\[1em] \Rightarrow 10x^2 + 15xy - 12xy - 18y^2 - 14xz - 21yz \\[1em] \Rightarrow 10x^2 + 3xy - 14xz - 18y^2 - 21yz.$

Hence, $\bm{(5x - 6y - 7z)(2x + 3y) = 10x^2 + 3xy - 14xz - 18y^2 - 21yz}$.

Find the product of:

$5x - 6y - 7z$ and $2x + 3y + z$

Answer

Solving,

$\Rightarrow (5x - 6y - 7z)(2x + 3y + z) \\[1em] \Rightarrow 5x(2x + 3y + z) - 6y(2x + 3y + z) - 7z(2x + 3y + z) \\[1em] \Rightarrow 10x^2 + 15xy + 5zx - 12xy - 18y^2 - 6yz - 14zx - 21yz - 7z^2 \\[1em] \Rightarrow 10x^2 + 3xy - 9zx - 18y^2 - 27yz - 7z^2.$

Hence, $\bm{(5x - 6y - 7z)(2x + 3y + z) = 10x^2 + 3xy - 9zx - 18y^2 - 27yz - 7z^2}$.

Find the product of:

$2a + 3b - 4c$ and $a - b - c$

Answer

Solving,

$\Rightarrow (2a + 3b - 4c)(a - b - c) \\[1em] \Rightarrow 2a(a - b - c) + 3b(a - b - c) - 4c(a - b - c) \\[1em] \Rightarrow 2a^2 - 2ab - 2ac + 3ab - 3b^2 - 3bc - 4ac + 4bc + 4c^2 \\[1em] \Rightarrow 2a^2 + ab - 6ac - 3b^2 + bc + 4c^2.$

Hence, $\bm{(2a + 3b - 4c)(a - b - c) = 2a^2 + ab - 6ac - 3b^2 + bc + 4c^2}$.