RS Aggarwal 2019,2020 solution class 7 chapter 6 Algebraic Expressions Exercise 6D

Exercise 6D

Page-106

Question 1:

Find the product:

(5x + 7)(3x + 4)

Answer 1:

$$\begin{gathered} (5\mathrm{x}+7)(3\mathrm{x}+4) \\ =5\mathrm{x}\times (3\mathrm{x}+4)+7\times (3\mathrm{x}+4) \\ =(5\mathrm{x}\times 3\mathrm{x} +5\mathrm{x}\times 4) +(7\times 3\mathrm{x}+7\times 4) \\ =15{\mathrm{x}}^2+20\mathrm{x} +21\mathrm{x} +28 \\ =15{\mathrm{x}}^2+41\mathrm{x}+28 \end{gathered}$$

Question 2:

Find the product:

(4x − 3)(2x + 5)

Answer 2:

$$\begin{gathered} (4\mathrm{x}-3)(2\mathrm{x}+5) \\ =4\mathrm{x}\times (2\mathrm{x}+5) -3\times (2\mathrm{x}+5) \\ =(4\mathrm{x}\times 2\mathrm{x} +4\mathrm{x}\times 5) -(3\times 2\mathrm{x} +3\times 5) \\ =8{\mathrm{x}}^2 +20\mathrm{x} -6\mathrm{x}-15 \\ =8{\mathrm{x}}^2+14\mathrm{x} -15 \end{gathered}$$

Question 3:

Find the product:

(x − 6)(4x + 9)

Answer 3:

$$\begin{gathered} (\mathrm{x}-6)(4\mathrm{x}+9) \\ =\mathrm{x}\times (4\mathrm{x}+9) -6\times (4\mathrm{x}+9) \\ =(\mathrm{x}\times 4\mathrm{x} +\mathrm{x}\times 9) -(6\times 4\mathrm{x} +6\times 9) \\ =4{\mathrm{x}}^2 +9\mathrm{x} -24\mathrm{x}-54 \\ =4{\mathrm{x}}^2-15\mathrm{x} -54 \end{gathered}$$

Question 4:

Find the product:

(5y − 1)(3y − 8)

Answer 4:

$$\begin{gathered} (5\mathrm{y}-1)(3\mathrm{y}-8) \\ =5\mathrm{y}\times (3\mathrm{y}-8) -1\times (3\mathrm{y}-8) \\ =(5\mathrm{y}\times 3\mathrm{y} -5\mathrm{y}\times 8) -(3\mathrm{y}-8) \\ =15{\mathrm{y}}^2-40\mathrm{y} -3\mathrm{y} +8 \\ =15{\mathrm{y}}^2-43\mathrm{y} +8 \end{gathered}$$

Question 5:

Find the product:

(7x + 2y)(x + 4y)

Answer 5:

$$\begin{gathered} (7\mathrm{x}+2\mathrm{y})(\mathrm{x}+4\mathrm{y}) \\ =7\mathrm{x}\times (\mathrm{x}+4\mathrm{y}) +2\mathrm{y}\times (\mathrm{x}+4\mathrm{y}) \\ =(7\mathrm{x}\times \mathrm{x} +7\mathrm{x}\times 4\mathrm{y}) +(2\mathrm{y}\times \mathrm{x}+2\mathrm{y}\times 4\mathrm{y}) \\ =7{\mathrm{x}}^2+28\mathrm{xy} +2\mathrm{xy} +8{\mathrm{y}}^2 \\ =7{\mathrm{x}}^2+30\mathrm{xy} +8{\mathrm{y}}^2 \end{gathered}$$

Question 6:

Find the product:

(9x + 5y)(4x + 3y)

Answer 6:

$$\begin{gathered} (9\mathrm{x}+5\mathrm{y})(4\mathrm{x}+3\mathrm{y}) \\ =9\mathrm{x}\times (4\mathrm{x}+3\mathrm{y}) +5\mathrm{y}\times (4\mathrm{x}+3\mathrm{y}) \\ =(9\mathrm{x}\times 4\mathrm{x} +9\mathrm{x}\times 3\mathrm{y}) +(5\mathrm{y}\times 4\mathrm{x} +5\mathrm{y}\times 3\mathrm{y}) \\ =36{\mathrm{x}}^2+27\mathrm{xy}+20\mathrm{xy}+15{\mathrm{y}}^2 \\ =36{\mathrm{x}}^2+47\mathrm{xy} +15{\mathrm{y}}^2 \end{gathered}$$

Question 7:

Find the product:

(3m − 4n)(2m − 3n)

Answer 7:

$$\begin{gathered} (3\mathrm{m}-4\mathrm{n})(2\mathrm{m}-3\mathrm{n}) \\ =3\mathrm{m}\times (2\mathrm{m}-3\mathrm{n}) -4\mathrm{n}\times (2\mathrm{m}-3\mathrm{n}) \\ =(3\mathrm{m}\times 2\mathrm{m} -3\mathrm{m}\times 3\mathrm{n}) -(4\mathrm{n}\times 2\mathrm{m} -4\mathrm{n}\times 3\mathrm{n}) \\ =6{\mathrm{m}}^2-9\mathrm{mn} -8\mathrm{nm} +12{\mathrm{n}}^2 \\ =6{\mathrm{m}}^2-17\mathrm{mn} +12{\mathrm{n}}^2 \end{gathered}$$

Question 8:

Find the product:

(0.8x −  0.5y)(1.5x −  3y)

Answer 8:

$$\begin{gathered} (0.8\mathrm{x}-0.5\mathrm{y})(1.5\mathrm{x}-3\mathrm{y}) \\ =0.8\mathrm{x}\times (1.5\mathrm{x}-3\mathrm{y}) -0.5\mathrm{y}\times (1.5\mathrm{x}-3\mathrm{y}) \\ =(0.8\mathrm{x}\times 1.5\mathrm{x} -0.8\mathrm{x}\times 3\mathrm{y}) -(0.5\mathrm{y}\times 1.5\mathrm{x} -0.5\mathrm{y}\times 3\mathrm{y}) \\ =1.2{\mathrm{x}}^2-2.4\mathrm{xy} -0.75\mathrm{yx} +1.5{\mathrm{y}}^2 \\ =1.2{\mathrm{x}}^2-3.15\mathrm{xy} +1.5{\mathrm{y}}^2 \end{gathered}$$

Question 9:

Find the product:

$\left(\frac{1}{5}x+2y\right)\left(\frac{2}{3}x-y\right)$

Answer 9:

$$\begin{gathered} \left(\frac{1}{5}\mathrm{x}+2\mathrm{y}\right)\left(\frac{2}{3}\mathrm{x}-\mathrm{y}\right) \\ =\frac{1}{5}\mathrm{x}\times \left(\frac{2}{3}\mathrm{x} -\mathrm{y}\right) +2\mathrm{y}\times \left(\frac{2}{3}\mathrm{x}-\mathrm{y}\right) \\ =\left(\frac{1}{5}\mathrm{x}\times \frac{2}{3}\mathrm{x} -\frac{1}{5}\mathrm{x}\times \mathrm{y}\right) +\left(2\mathrm{y}\times \frac{2}{3}\mathrm{x} -2\mathrm{y}\times \mathrm{y}\right) \\ =\frac{2}{15}{\mathrm{x}}^2-\frac{1}{5}\mathrm{xy} +\frac{4}{3}\mathrm{yx}-2{\mathrm{y}}^2 \\ =\frac{2}{15}{\mathrm{x}}^2+\frac{17}{15}\mathrm{xy} -2{\mathrm{y}}^2 \end{gathered}$$

Question 10:

Find the product:

$\left(\frac{2}{5}x-\frac{1}{2}y\right)(10x-8y)$

Answer 10:

$$\begin{gathered} (\frac{2}{5}\mathrm{x}-\frac{1}{2}\mathrm{y})(10\mathrm{x}-8\mathrm{y}) \\ =\frac{2}{5}\mathrm{x}\times (10\mathrm{x}-8\mathrm{y}) -\frac{1}{2}\mathrm{y}\times (10\mathrm{x} -8\mathrm{y}) \\ =(\frac{2}{5}\mathrm{x}\times 10\mathrm{x} - \frac{2}{5}\mathrm{x}\times 8\mathrm{y}) -(\frac{1}{2}\mathrm{y} \times 10\mathrm{x} -\frac{1}{2}\mathrm{y}\times 8\mathrm{y}) \\ =4{\mathrm{x}}^2-\frac{16}{5}\mathrm{xy} -5\mathrm{yx} +4{\mathrm{y}}^2 \\ =4{\mathrm{x}}^2 -\frac{41}{5}\mathrm{xy} +4{\mathrm{y}}^2 \end{gathered}$$

Question 11:

Find the product:

$\left(\frac{3}{4}a+\frac{2}{3}b\right)(4a+3b)$

Answer 11:

$$\begin{gathered} (\frac{3}{4}\mathrm{a}+\frac{2}{3}\mathrm{b})(4\mathrm{a}+3\mathrm{b}) \\ =\frac{3}{4}\mathrm{a}\times (4\mathrm{a}+3\mathrm{b}) +\frac{2}{3}\mathrm{b}\times (4\mathrm{a}+3\mathrm{b}) \\ =(\frac{3}{4}\mathrm{a}\times 4\mathrm{a} +\frac{3}{4}\mathrm{a}\times 3\mathrm{b})+(\frac{2}{3}\mathrm{b}\times 4\mathrm{a} +\frac{2}{3}\mathrm{b}\times 3\mathrm{b}) \\ =3{\mathrm{a}}^2 +\frac{9}{4}\mathrm{ab}+\frac{8}{3}\mathrm{ba} +2{\mathrm{b}}^2 \\ =3{\mathrm{a}}^2+\frac{59}{12}\mathrm{ab} +2{\mathrm{b}}^2 \end{gathered}$$

Question 12:

Find the product:

(x² − a²)(x −a)

Answer 12:

$$\begin{gathered} ({\mathrm{x}}^2-{\mathrm{a}}^2)(\mathrm{x}-\mathrm{a}) \\ ={\mathrm{x}}^2\times (\mathrm{x}-\mathrm{a}) -{\mathrm{a}}^2\times (\mathrm{x}-\mathrm{a}) \\ =({\mathrm{x}}^2\times \mathrm{x} -{\mathrm{x}}^2\times \mathrm{a}) -({\mathrm{a}}^2\times \mathrm{x} -{\mathrm{a}}^2\times \mathrm{a}) \\ ={\mathrm{x}}^{3 }-{\mathrm{ax}}^2 -{\mathrm{a}}^2\mathrm{x} +{\mathrm{a}}^3 \end{gathered}$$

Question 13:

Find the product:

(3p² + q²)(2p² − 3q²)

Answer 13:

$$\begin{gathered} (3{\mathrm{p}}^2+{\mathrm{q}}^2(2{\mathrm{p}}^2-3{\mathrm{q}}^2) \\ =3{\mathrm{p}}^2\times (2{\mathrm{p}}^2-3{\mathrm{q}}^2) +{\mathrm{q}}^2\times (2{\mathrm{p}}^2-3{\mathrm{q}}^2) \\ =(3{\mathrm{p}}^2\times 2{\mathrm{p}}^2-3{\mathrm{p}}^2\times 3{\mathrm{q}}^2) +({\mathrm{q}}^2\times 2{\mathrm{p}}^2 -{\mathrm{q}}^2\times 3{\mathrm{q}}^2) \\ =6{\mathrm{p}}^4-9{\mathrm{p}}^2{\mathrm{q}}^2 +2{\mathrm{q}}^2{\mathrm{p}}^2 -3{\mathrm{q}}^4 \\ =6{\mathrm{p}}^4 -7{\mathrm{p}}^2{\mathrm{q}}^2-3{\mathrm{q}}^4 \end{gathered}$$

Question 14:

Find the product:

(2x² − 5y²)(x² + 3y²)

Answer 14:

$$\begin{gathered} (2{\mathrm{x}}^2-5{\mathrm{y}}^2)({\mathrm{x}}^2 +3{\mathrm{y}}^2) \\ =2{\mathrm{x}}^2\times ({\mathrm{x}}^2 +3{\mathrm{y}}^2) -5{\mathrm{y}}^2\times ({\mathrm{x}}^2 +3{\mathrm{y}}^2) \\ =(2{\mathrm{x}}^2\times {\mathrm{x}}^2 +2{\mathrm{x}}^2\times 3{\mathrm{y}}^2) -(5{\mathrm{y}}^2\times {\mathrm{x}}^2 +5{\mathrm{y}}^2\times 3{\mathrm{y}}^2) \\ =2{\mathrm{x}}^4 +6{\mathrm{x}}^2{\mathrm{y}}^2 -5{\mathrm{y}}^2{\mathrm{x}}^2 -15{\mathrm{y}}^4 \\ =2{\mathrm{x}}^4 +{\mathrm{x}}^2{\mathrm{y}}^2 -15{\mathrm{y}}^4 \end{gathered}$$

Question 15:

Find the product:

(x³ − y³)(x² + y²)

Answer 15:

$$\begin{gathered} ({\mathrm{x}}^3-{\mathrm{y}}^3)({\mathrm{x}}^2+{\mathrm{y}}^2) \\ ={\mathrm{x}}^3\times ({\mathrm{x}}^2 +{\mathrm{y}}^2) -{\mathrm{y}}^3\times ({\mathrm{x}}^2+{\mathrm{y}}^2) \\ =({\mathrm{x}}^3\times {\mathrm{x}}^2+{\mathrm{x}}^3\times {\mathrm{y}}^2) -({\mathrm{y}}^3\times {\mathrm{x}}^2 +{\mathrm{y}}^3\times {\mathrm{y}}^2) \\ ={\mathrm{x}}^5 +{\mathrm{x}}^3{\mathrm{y}}^2-{\mathrm{x}}^2{\mathrm{y}}^3 -{\mathrm{y}}^5 \end{gathered}$$

Question 16:

Find the product:

(x⁴ + y⁴)(x² − y²)

Answer 16:

$$\begin{gathered} ({\mathrm{x}}^4+{\mathrm{y}}^4)({\mathrm{x}}^2 -{\mathrm{y}}^2) \\ ={\mathrm{x}}^4\times ({\mathrm{x}}^2 -{\mathrm{y}}^2) +{\mathrm{y}}^4\times ({\mathrm{x}}^2-{\mathrm{y}}^2) \\ =({\mathrm{x}}^4\times {\mathrm{x}}^2-{\mathrm{x}}^4\times {\mathrm{y}}^2) +({\mathrm{y}}^4\times {\mathrm{x}}^2 -{\mathrm{y}}^4\times {\mathrm{y}}^2) \\ ={\mathrm{x}}^6-{\mathrm{x}}^4{\mathrm{y}}^2+{\mathrm{x}}^2{\mathrm{y}}^4 -{\mathrm{y}}^6 \end{gathered}$$

Question 17:

Find the product:

$\left(x^4+\frac{1}{x^4}\right)\left(x+\frac{1}{x}\right)$

Answer 17:

$$\begin{gathered} \left({\mathrm{x}}^4+\frac{1}{{\mathrm{x}}^4}\right)(\mathrm{x}+\frac{1}{\mathrm{x}}) \\ ={\mathrm{x}}^4\times (\mathrm{x}+\frac{1}{\mathrm{x}}) +\frac{1}{{\mathrm{x}}^4}\times (\mathrm{x}+\frac{1}{\mathrm{x}}) \\ =({\mathrm{x}}^4\times \mathrm{x} + {\mathrm{x}}^4\times \frac{1}{\mathrm{x}})+(\frac{1}{{\mathrm{x}}^4}\times \mathrm{x} +\frac{1}{{\mathrm{x}}^4}\times \frac{1}{\mathrm{x}}) \\ ={\mathrm{x}}^5+{\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}+\frac{1}{{\mathrm{x}}^5} \end{gathered}$$

Question 18:

Find the product:

(x² − y²)(x + 2y)

Answer 18:

$$\begin{gathered} ({\mathrm{x}}^2-{\mathrm{y}}^2)(\mathrm{x}+2\mathrm{y}) \\ ={\mathrm{x}}^2\times (\mathrm{x}+2\mathrm{y}) -{\mathrm{y}}^2\times (\mathrm{x}+2\mathrm{y}) \\ =({\mathrm{x}}^2\times \mathrm{x} +{\mathrm{x}}^2\times 2\mathrm{y}) -({\mathrm{y}}^2\times \mathrm{x} +{\mathrm{y}}^2\times 2\mathrm{y}) \\ ={\mathrm{x}}^3 +2{\mathrm{x}}^2\mathrm{y}-{\mathrm{xy}}^2 -2{\mathrm{y}}^3 \end{gathered}$$

Question 19:

Find the product:

(2x + 3y − 5)(x + y)

Answer 19:

$$\begin{gathered} (2\mathrm{x}+3\mathrm{y}-5)(\mathrm{x}+\mathrm{y}) \\ =2\mathrm{x}\times (\mathrm{x}+\mathrm{y}) +3\mathrm{y}\times (\mathrm{x}+\mathrm{y}) -5\times (\mathrm{x}+\mathrm{y}) \\ =(2\mathrm{x}\times \mathrm{x} +2\mathrm{x}\times \mathrm{y} )+(3\mathrm{y}\times \mathrm{x} +3\mathrm{y}\times \mathrm{y}) -(5\times \mathrm{x} +5\times \mathrm{y}) \\ =2{\mathrm{x}}^2 +2\mathrm{x}\mathrm{y} +3\mathrm{y}\mathrm{x} +3{\mathrm{y}}^2-5\mathrm{x} -5\mathrm{y} \\ =2{\mathrm{x}}^2 +5\mathrm{x}\mathrm{y} -5\mathrm{x} -5\mathrm{y}+3{\mathrm{y}}^2 \end{gathered}$$

Question 20:

Find the product:

(3x + 2y − 4)(x − y)

Answer 20:

By column method:

$$\begin{gathered} 3\mathrm{x} +2\mathrm{y} -4 \\ \underline{ \times ( \mathrm{x}- \mathrm{y})} \\ 3{\mathrm{x}}^2 +2\mathrm{yx} -4\mathrm{x} \\ \underline{ -3\mathrm{xy} -2{\mathrm{y}}^2+4\mathrm{y}} \\ \mathrm{Add}: 3{\mathrm{x}}^2 -\mathrm{xy} -4\mathrm{x} +4\mathrm{y} -2{\mathrm{y}}^2 \\ \therefore (3\mathrm{x} +2\mathrm{y} -4)(\mathrm{x}-\mathrm{y}) = 3{\mathrm{x}}^2 -\mathrm{xy} -4\mathrm{x} +4\mathrm{y} -2{\mathrm{y}}^2 \end{gathered}$$

Question 21:

Find the product:

(x² − 3x + 7)(2x + 3)

Answer 21:

By column method:

$$\begin{gathered} {\mathrm{x}}^2-3\mathrm{x} +7 \\ \underline{ \times (2\mathrm{x}+3)} \\ 2{\mathrm{x}}^3 -6{\mathrm{x}}^2+14\mathrm{x} \\ \underline{ 3{\mathrm{x}}^2 -9\mathrm{x} +21} \\ \mathrm{Add}: 2{\mathrm{x}}^3 -3{\mathrm{x}}^2 +5\mathrm{x} +21 \\ \therefore ({\mathrm{x}}^2-3\mathrm{x} +7)(2\mathrm{x}+3) = 2{\mathrm{x}}^3 -3{\mathrm{x}}^2 +5\mathrm{x} +21 \end{gathered}$$

Question 22:

Find the product:

(3x² + 5x − 9)(3x −9)

Answer 22:

By column method:

$$\begin{gathered} 3{\mathrm{x}}^2 +5\mathrm{x} -9 \\ \underline{ \times (3\mathrm{x} -9)} \\ 9{\mathrm{x}}^3 +15{\mathrm{x}}^2 -27\mathrm{x} \\ \underline{ -27{\mathrm{x}}^2 -45\mathrm{x} +81 } \\ \mathrm{Add}: 9{\mathrm{x}}^3 -12{\mathrm{x}}^2 -72\mathrm{x} +81 \\ \therefore (3{\mathrm{x}}^2 +5\mathrm{x} -9)(3\mathrm{x}-9) =9{\mathrm{x}}^3 -12{\mathrm{x}}^2 -72\mathrm{x} +81 \end{gathered}$$

Question 23:

Find the product:

(9x² − x + 15)(x² − 3)

Answer 23:

By column method:

$$\begin{gathered} 9{\mathrm{x}}^2 -\mathrm{x} +15 \\ \underline{ \times ({\mathrm{x}}^2 -3)} \\ 9{\mathrm{x}}^4-{\mathrm{x}}^3 +15{\mathrm{x}}^2 \\ \underline{ -27{\mathrm{x}}^2 +3\mathrm{x} -45 } \\ \mathrm{Add}: 9{\mathrm{x}}^4-{\mathrm{x}}^3-12{\mathrm{x}}^2+3\mathrm{x} -45 \\ \therefore (9{\mathrm{x}}^2 -\mathrm{x} +15)({\mathrm{x}}^2 -3) = 9{\mathrm{x}}^4-{\mathrm{x}}^3-12{\mathrm{x}}^2+3\mathrm{x} -45 \end{gathered}$$

Question 24:

Find the product:

(x² + xy + y²)(x − y)

Answer 24:

By column method:

$$\begin{gathered} {\mathrm{x}}^2+\mathrm{xy} +{\mathrm{y}}^2 \\ \underline{ \times (\mathrm{x}-\mathrm{y})} \\ {\mathrm{x}}^3+{\mathrm{x}}^2\mathrm{y} +{\mathrm{xy}}^2 \\ \underline{ -{\mathrm{x}}^2\mathrm{y} -{\mathrm{xy}}^2 -{\mathrm{y}}^3 } \\ \mathrm{Add}: {\mathrm{x}}^3 -{\mathrm{y}}^3 \\ \therefore ( {\mathrm{x}}^2+\mathrm{xy} +{\mathrm{y}}^2 )(\mathrm{x}-\mathrm{y}) ={\mathrm{x}}^3 -{\mathrm{y}}^3 \end{gathered}$$

Question 25:

Find the product:

(x² − xy + y²)(x + y)

Answer 25:

By column method:

$$\begin{gathered} {\mathrm{x}}^2 -\mathrm{xy}+{\mathrm{y}}^2 \\ \underline{ \times (\mathrm{x}+\mathrm{y})} \\ {\mathrm{x}}^3-{\mathrm{x}}^2\mathrm{y} +{\mathrm{xy}}^2 \\ \underline{ {\mathrm{x}}^2\mathrm{y} -{\mathrm{xy}}^2 +{\mathrm{y}}^3} \\ \mathrm{Add}: {\mathrm{x}}^3 +{\mathrm{y}}^3 \\ \therefore ({\mathrm{x}}^2 -\mathrm{xy}+{\mathrm{y}}^2)(\mathrm{x} + \mathrm{y}) ={\mathrm{x}}^3 +{\mathrm{y}}^3 \end{gathered}$$

Question 26:

Find the product:

(x² − 5x + 8)(x² + 2)

Answer 26:

By column method:
$$\begin{gathered} {\mathrm{x}}^2-5\mathrm{x} +8 \\ \underline{ \times ({\mathrm{x}}^2 +2)} \\ {\mathrm{x}}^4 -5{\mathrm{x}}^3 +8{\mathrm{x}}^{2 } \\ \underline{ 2{\mathrm{x}}^2 -10\mathrm{x} +16 } \\ \mathrm{Add}: {\mathrm{x}}^4 -5{\mathrm{x}}^3 +10{\mathrm{x}}^2 -10\mathrm{x} +16 \\ \therefore ({\mathrm{x}}^2-5\mathrm{x} +8)({\mathrm{x}}^2+2) = {\mathrm{x}}^4 -5{\mathrm{x}}^3 +10{\mathrm{x}}^2 -10\mathrm{x} +16 \end{gathered}$$

Question 27:

Simplify

(3x + 4)(2x − 3) + (5x − 4)(x + 2)

Answer 27:

(3x +4)(2x -3)
$$\begin{gathered} =3\mathrm{x}\times (2\mathrm{x} -3) +4\times (2\mathrm{x}-3) \\ =6{\mathrm{x}}^{(1+1)} -9\mathrm{x} +8\mathrm{x} -12 \\ =6{\mathrm{x}}^2 -\mathrm{x} -12 \\ (5\mathrm{x}-4)(\mathrm{x}+2) \\ =5\mathrm{x}(\mathrm{x}+2) -4(\mathrm{x}+2) \\ =5{\mathrm{x}}^{(1+1)} +10\mathrm{x} -4\mathrm{x} -8 \\ =5{\mathrm{x}}^2 +6\mathrm{x} -8 \end{gathered}$$

∴ (3x + 4)(2x − 3) + (5x − 4)(x + 2)

$$\begin{gathered} =6x^2-x-12 +5x^2 +6x -8 \\ =11x^2 +5x -20 \end{gathered}$$

Question 28:

Simplify

(5x − 3)(x + 4) − (2x + 5)(3x − 4)

Answer 28:

(5x-3)(x+4)

$$\begin{gathered} =5\mathrm{x}\times (\mathrm{x}+4) -3\times (\mathrm{x}+4) \\ =5{\mathrm{x}}^{(1+1)} +20\mathrm{x} -3\mathrm{x} -12 \\ =5{\mathrm{x}}^2+17\mathrm{x} -12 \end{gathered}$$

(2x +5)(3x-4)

$$\begin{gathered} =2\mathrm{x}\times (3\mathrm{x}-4) +5\times (3\mathrm{x}-4) \\ =6{\mathrm{x}}^{(1+1)} -8\mathrm{x} +15\mathrm{x} -20 \\ =6{\mathrm{x}}^2+7\mathrm{x} -20 \end{gathered}$$

∴ (5x − 3)(x + 4) − (2x + 5)(3x − 4)

$$\begin{gathered} =5{\mathrm{x}}^2+17\mathrm{x} -12 -(6{\mathrm{x}}^2 +7\mathrm{x} -20) \\ =5{\mathrm{x}}^2 -6{\mathrm{x}}^2 +17\mathrm{x} -7\mathrm{x} -12 +20 \\ =-{\mathrm{x}}^2 +10\mathrm{x} +8 \end{gathered}$$

Question 29:

Simplify

(9x − 7)(2x − 5) − (3x − 8)(5x − 3)

Answer 29:

$$\begin{gathered} (9\mathrm{x}-7)(2\mathrm{x}-5) \\ =9\mathrm{x}\times (2\mathrm{x}-5) -7\times (2\mathrm{x}-5) \\ =18{\mathrm{x}}^{(1+1)} -45\mathrm{x} -14\mathrm{x} +35 \\ =18{\mathrm{x}}^2 -59\mathrm{x} +35 \\ (3\mathrm{x}-8)(5\mathrm{x}-3) \\ =3\mathrm{x}\times (5\mathrm{x}-3) -8\times (5\mathrm{x}-3) \\ =15{\mathrm{x}}^2 -9\mathrm{x} -40\mathrm{x} +24 \\ =15{\mathrm{x}}^2-49\mathrm{x} +24 \\ \therefore (2\mathrm{x\; -\; 5)\; -\; (3x\; -\; 8)(5x\; -\; 3)} \\ =18{\mathrm{x}}^2 -59\mathrm{x} +35 -(15{\mathrm{x}}^2-49\mathrm{x} +24) \\ =18{\mathrm{x}}^2-15{\mathrm{x}}^2 -59\mathrm{x} +49\mathrm{x} +35 -24 \\ =3{\mathrm{x}}^2 -10\mathrm{x} +11 \end{gathered}$$

Question 30:

Simplify

(2x + 5y)(3x + 4y) − (7x + 3y)(2x − y)

Answer 30:

(2x +5y)(3x+4y)

$$\begin{gathered} =2\mathrm{x}\times (3\mathrm{x}+4\mathrm{y}) +5\mathrm{y}\times (3\mathrm{x}+4\mathrm{y}) \\ =6{\mathrm{x}}^{(1+1)} +8\mathrm{xy} +15\mathrm{yx} +20{\mathrm{y}}^{(1+1)} \\ =6{\mathrm{x}}^2 +23\mathrm{xy} +20{\mathrm{y}}^2 \\ (7\mathrm{x} +3\mathrm{y})(2\mathrm{x} +\mathrm{y}) \\ =7\mathrm{x}(2\mathrm{x} +\mathrm{y}) +3\mathrm{y}(2\mathrm{x} +\mathrm{y}) \\ =14{\mathrm{x}}^{(1+1)} +7\mathrm{xy} +6\mathrm{yx} +3{\mathrm{y}}^{(1+1)} \\ =14{\mathrm{x}}^2 +13\mathrm{xy}+3{\mathrm{y}}^2 \end{gathered}$$

∴ (2x + 5y)(3x + 4y) − (7x + 3y)(2x − y)

$$\begin{gathered} =6{\mathrm{x}}^2+23\mathrm{xy} +20{\mathrm{y}}^2 -(14{\mathrm{x}}^2 +13\mathrm{xy} +3{\mathrm{y}}^2) \\ =6{\mathrm{x}}^2 -14{\mathrm{x}}^2 +23\mathrm{xy} -13\mathrm{xy} +20{\mathrm{y}}^2 -3{\mathrm{y}}^2 \\ =-8{\mathrm{x}}^2 +10\mathrm{xy} +17{\mathrm{y}}^2 \end{gathered}$$

Question 31:

Simplify

(3x² + 5x − 7)(x − 1) − (x² − 2x + 3)(x + 4)

Answer 31:

(3x² + 5x − 7)(x − 1)

By column method:

$$\begin{gathered} 3{\mathrm{x}}^2 +5\mathrm{x} -7 \\ \underline{ \times (\mathrm{x} -1)} \\ 3{\mathrm{x}}^3 +5{\mathrm{x}}^2 -7\mathrm{x} \\ \underline{ -3{\mathrm{x}}^2 -5\mathrm{x} +7 } \\ \mathrm{Add}: 3{\mathrm{x}}^3 +2{\mathrm{x}}^2 -12\mathrm{x}+7 \end{gathered}$$


(x² − 2x + 3)(x + 4)

By column method:

$$\begin{gathered} {\mathrm{x}}^2 -2\mathrm{x} +3 \\ \underline{ \times (\mathrm{x} +4)} \\ {\mathrm{x}}^3 -2{\mathrm{x}}^2 +3\mathrm{x} \\ \underline{ 4{\mathrm{x}}^2 -8\mathrm{x} +12 } \\ \mathrm{Add}: {\mathrm{x}}^3 +2{\mathrm{x}}^2 -5\mathrm{x} +12 \end{gathered}$$

(3x² + 5x − 7)(x − 1) − (x² − 2x + 3)(x + 4)

$$\begin{gathered} =3{\mathrm{x}}^3 +2{\mathrm{x}}^2 -12\mathrm{x} +7 -({\mathrm{x}}^3+2{\mathrm{x}}^2 -5\mathrm{x} +12) \\ =3{\mathrm{x}}^3-{\mathrm{x}}^3 +2{\mathrm{x}}^2 -2{\mathrm{x}}^2 -12\mathrm{x} +5\mathrm{x} +7 -12 \\ =2{\mathrm{x}}^3 -7\mathrm{x} -5 \end{gathered}$$