Solution

\begin{align*} &= (x + 3)(x^2 + 2x - 1) \\ &= x(x^2 + 2x - 1) + 3(x^2 + 2x - 1) \\ &= x^3 + 2x^2 - x + 3x^2 + 6x - 3 \\ &= x^3 {\color{magenta} + 2x^2 + 3x^2} {\color{green} -x + 6x} - 3 \\ &= x^3 {\color{magenta} + 5x^2} {\color{green} + 5x} - 3 \end{align*}

Answer \( \color{red} x^3 + 5x^2 + 5x - 3 \)

Solution

\begin{align*} &= (7y - 2)(5y^2 - 3y + 2) \\ &= 7y(5y^2 - 3y + 2) - 2(5y^2 - 3y + 2) \\ &= 35y^3 - 21y^2 + 14y - 10y^2 + 6y - 4 \\ &= 35y^3 { \color{magenta} - 21y^2 - 10y^2} {\color{green} + 14y + 6y} - 4 \\ &= 35y^3 {\color{magenta} - 31y^2} {\color{green} + 20y} - 4 \end{align*}

Answer \( \color{red} 35y^3 - 31y^2 + 20y - 4 \)

Solution

\begin{align*} &= (p^3 + 3p + q)(9p + 2q) \\ &= p^3(9p + 2q) + 3p(9p + 2q) + q(9p + 2q) \\ &= 9p^4 + 2p^3q + 27p^2 {\color{magenta} + 6pq + 9pq }+ 2q^2 \\ &= 9p^4 + 2p^3q + 27p^2 {\color{magenta} + 15pq} + 2q^2 \end{align*}

Answer \( \color{red} 9p^4 + 2p^3q + 27p^2 + 15pq + 2q^2 \)

Solution

\begin{align*} &= (-2x^2 + xy - y^2)(3x + 4y) \\ &= -2x^2(3x + 4y) + xy(3x + 4y) - y^2(3x + 4y) \\ &= -6x^3 {\color{magenta} - 8x^2y + 3x^2y } { \color{green} + 4xy^2 - 3xy^2} - 4y^3 \\ &= -6x^3 {\color{magenta} - 5x^2y} {\color{green} + xy^2 }- 4y^3 \end{align*}

Answer \( \color{red} -6x^3 - 5x^2y + xy^2 - 4y^3 \)

Solution

\begin{align*} &= \left(\frac{2}{5} a + \frac{1}{7}b\right) (3a + 4b - 2) \\ \\ &= \frac{2}{5} a (3a + 4b - 2) + \frac{1}{7}b (3a + 4b - 2) \\ \\ &= \frac{6}{5} a^2 + \frac{8}{5} ab - \frac{4}{5} a + \frac{3}{7} ab + \frac{4}{7} b^2 - \frac{2}{7}b \\ \\ &= \frac{6}{5} a^2 {\color{magenta} + \frac{8}{5} ab + \frac{3}{7} ab} - \frac{4}{5} a + \frac{4}{7} b^2 - \frac{2}{7}b \\ \\ &= \frac{6}{5} a^2 {\color{magenta} + \frac{71}{35} ab} - \frac{4}{5} a + \frac{4}{7} b^2 - \frac{2}{7}b \\ \end{align*}

Answer \( \color{red} \frac{6}{5} a^2 + \frac{71}{35} ab - \frac{4}{5} a + \frac{4}{7} b^2 - \frac{2}{7}b \)

Solution

\begin{align*} &= (0.1a - 0.2c) (a + c + ac) \\ &= 0.1a(a + c + ac) - 0.2c(a + c + ac) \\ &= 0.1a^2 + 0.1ac + 0.1a^2c - 0.2ac - 0.2c^2 - 0.2ac^2 \\ &= 0.1a^2 {\color{magenta} + 0.1ac - 0.2ac} + 0.1a^2c - 0.2c^2 - 0.2ac^2 \\ &= 0.1a^2 {\color{magenta} - 0.1ac }+ 0.1a^2c - 0.2c^2 - 0.2ac^2 \end{align*}

Answer \( \color{red} 0.1a^2 - 0.1ac + 0.1a^2c - 0.2c^2 - 0.2ac^2 \)

Solution

\begin{align*} &=(x^2 - 4xy + y^2)(x - 2y) \\ &= x^2 (x - 2y) - 4xy (x - 2y) + y^2 (x - 2y) \\ &= x^3 - 2x^2y - 4x^2y + 8xy^2 + xy^2 - 2y^3 \\ &= x^3 - 6x^2y + 9xy^2 - 2y^3 \\ \\ & {\color{magenta} \text{Verification: }} x = 3 , y = 2 \\ & \color{magenta} \text{L.H.S} \\ &=(x^2 - 4xy + y^2) \times (x - 2y) \\ &=[(3)^2 - 4(3) (2) + (2)^2] \times [3 - 2 (2)] \\ &=(9 - 24 + 4) \times (3 - 4) \\ &=-11 \times (-1) \\ &= 11 \\ \\ & \color{magenta} \text{R.H.S} \\ &= x^3 - 6x^2y + 9xy^2 - 2y^3 \\ &= (3)^3 - 6 (3)^2 (2) + 9 (3)(2)^2 - 2(2)^3 \\ &= 27 - (6 \times 9 \times 2) + (9 \times 3 \times 4) - 2 \times 8 \\ &= 27 - \cancel{108} + \cancel{108} - 16 \\ &= 27 - 16 \\ &= 11 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution

\begin{align*} &=(7x^2y - 3z^2) (x + y + z) \\ &= 7x^2y(x + y + z) - 3z^2(x + y + z) \\ &= 7x^3y + 7x^2y^2 + 7x^2yz - 3xz^2 - 3yz^2 - 3z^3 \\ \\ & {\color{magenta} \text{Verification: }} x = 1, y = 1, z = -1 \\ & \color{magenta} \text{L.H.S} \\ &=(7x^2y - 3z^2) \times (x + y + z) \\ &=(7(1)^2(1) - 3(-1)^2) \times (1 + 1 - 1) \\ &=(7 - 3) \times 1 \\ &= 4 \\ \\ & \color{magenta} \text{R.H.S} \\ &= 7x^3y + 7x^2y^2 + 7x^2yz - 3xz^2 - 3yz^2 - 3z^3 \\ &= 7(1)^3(1) + 7(1)^2(1)^2 + 7(1)^2(1)(-1) - 3(1)(-1)^2 - 3(1)(-1)^2 - 3(-1)^3 \\ &= 7 + \cancel 7 - \cancel 7 - 3 - \cancel 3 + \cancel 3 \\ &= 7 - 3 \\ &= 4 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution

\begin{align*} &=(4a^2 - 6ab + 9b^2)(2a + 3b) \\ &= 4a^2(2a + 3b) - 6ab(2a + 3b) + 9b^2(2a + 3b) \\ &= 8a^3 + \cancel{12a^2b} - \cancel{12a^2b} - \cancel{18ab^2} + \cancel{18ab^2} + 27b^3 \\ &= 8a^3 + 27b^3 \\ \\ & {\color{magenta} \text{Verification: }} a = 2 , b = 1 \\ & \color{magenta} \text{L.H.S} \\ &= (4a^2 - 6ab + 9b^2)(2a + 3b) \\ &=[4(2)^2 - 6(2)(1) + 9(1)^2] \times [2(2) + 3(1)] \\ &=(16 - 12 + 9) \times (4 + 3) \\ &= 13 \times 7 \\ &= 91 \\ \\ & \color{magenta} \text{R.H.S} \\ &= 8a^3 + 27b^3 \\ &= 8(2)^3 + 27(1)^3 \\ &= 8 \times 8 + 27 \\ &= 64 + 27 \\ &= 91 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution

\begin{align*} &=(m^2 - 10m + 25) (m - 5) \\ &= m^2(m - 5) - 10m(m - 5) + 25(m - 5) \\ &= m^3 - 5m^2 - 10m^2 + 50m + 25m - 125 \\ &= m^3 - 15m^2 + 75m - 125 \\ \\ & {\color{magenta} \text{Verification: }} m = -2 \\ & \color{magenta} \text{L.H.S} \\ &=(m^2 - 10m + 25) \times (m - 5) \\ &=[(-2)^2 - 10(-2) + 25] \times (-2 - 5) \\ &=(4 + 20 + 25) \times (-7) \\ &= 49 \times (-7) \\ &= -343 \\ \\ & \color{magenta} \text{R.H.S} \\ &= m^3 - 15m^2 + 75m - 125 \\ &= (-2)^3 - 15(-2)^2 + 75(-2) - 125 \\ &= -8 - 60 - 150 - 125 \\ &= -343 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution

\begin{align*} &=(p^2 + q^2 + r^2)(pq + qr) \\ &= p^2(pq + qr) + q^2(pq + qr) + r^2(pq + qr) \\ &= p^3q + p^2qr + pq^3 + q^3r + pqr^2 + qr^3 \\ \\ & {\color{magenta} \text{Verification: }} p = 2, q = -3, r = 1 \\ & \color{magenta} \text{L.H.S} \\ &=(p^2 + q^2 + r^2) \times (pq + qr) \\ &= [(2)^2 + (-3)^2 + (1)^2] \times [2(-3) + 1(-3)] \\ &=(4 + 9 + 1) \times (-6 - 3) \\ &=14 \times (-9) \\ &=-126 \\ \\ & \color{magenta} \text{R.H.S} \\ &= p^3q + p^2qr + pq^3 + q^3r + pqr^2 + qr^3 \\ &= (2)^3(-3) + (2)^2(-3)(1) + 2(-3)^3 + (-3)^3(1) + (2)(-3)(1)^2 + (-3)(1)^3 \\ &= -24 - 12 - 54 - 27 - 6 - 3 \\ &= -126 \\ \\& \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution

\begin{align*} &=\left(\frac{5}{4} x^2 - \frac{3}{2} xy\right)(x + y + y^2) \\ \\ &= \frac{5}{4}x^2(x + y + y^2) - \frac{3}{2}xy(x + y + y^2) \\ \\ &= \frac{5}{4} x^3 + \frac{5}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} x^2y - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ &= \frac{5}{4} x^3 + \frac{5}{4} x^2y - \frac{3}{2} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ &= \frac{5}{4} x^3 - \frac{1}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ & {\color{magenta} \text{Verification: }} x = 2, y = 2 \\ & \color{magenta} \text{L.H.S} \\ &=\left(\frac{5}{4} x^2 - \frac{3}{2} xy\right)(x + y + y^2) \\ \\ &=\left[\frac{5}{4} (2)^2 - \frac{3}{2} (2)(2)\right] \times [2 + 2 + (2)^2] \\ &=\left(\frac{5}{\cancel4} \times \cancel4 - \frac{3}{\cancel2} \times \cancel4^2 \right) \times (2 + 2 + 4) \\ &=\left(5 - 6\right) \times 8 \\ &=(-1) \times 8 \\ &=-8 \\ \\& \color{magenta} \text{R.H.S} \\ &= \frac{5}{4} x^3 - \frac{1}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ &= \frac{5}{4} (2)^3 - \frac{1}{4} (2)^2(2) + \frac{5}{4} (2)^2(2)^2 - \frac{3}{2} (2)(2)^2 - \frac{3}{2} (2)(2)^3 \\ \\ &= \left(\frac{5}{\cancel4} \times \cancel{8}^2 \right)- \left(\frac{1}{\cancel4} \times \cancel4 \times 2 \right)+ \left(\frac{5}{\cancel4} \times \cancel4 \times 4 \right)- \left(\frac{3}{\cancel2} \times \cancel2 \times 4 \right)- \left(\frac{3}{\cancel2} \times \cancel2 \times 8\right) \\ \\ &= 10 - 2 + 20 - 12 - 24 \\ &= 30 - 38 \\ &= -8 \\ \\& \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}