RD Sharma Solutions Class 9 Factorization Of Algebraic Expressions Exercise 5.2

RD Sharma Solutions Class 9 Chapter 5 Ex 5.2

Q 1 . \(p^{3}+27\)

SOLUTION :

= \(p^{3}+3^{3}\)                                       \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

= (p + 3)(p² – 3p – 9)

\(∴\) \(p^{3}+27\) =(p + 3)(p² – 3p – 9)

Q 2 . \(y^{3}+125\)

SOLUTION :

=  \(y^{3}+5^{3}\)                                       \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

= \(\left ( y+5 \right )\left ( y^{2} -5y+5^{2}\right )\)

= \(\left ( y+5 \right )\left ( y^{2} -5y+25\right )\)

\(∴\) \(y^{3}+125\) = \(\left ( y+5 \right )\left ( y^{2} -5y+25\right )\)

Q 3 . \(1-27a^{3}\)

SOLUTION :

= \(\left ( 1 \right )^{3}-\left ( 3a \right )^{3}\)

= \(\left ( 1-3a \right )\left ( 1^{2}+1\times 3a+\left ( 3a \right )^{2} \right )\)                  \(∵ \left [ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2} +ab+b^{2}\right ) \right ]\)

= \(\left ( 1-3a \right )\left ( 1^{2}+3a+9a^{2}\right )\)

\(∴\) \(1-27a^{3}\) = \(\left ( 1-3a \right )\left ( 1^{2}+3a+9a^{2}\right )\)

Q 4 . \(8x^{3}y^{3}+27a^{3}\)

SOLUTION :

= \(\left ( 2xy \right )^{3}+\left ( 3a \right )^{3}\)

= \(\left ( 2xy+3a \right )\left ( \left ( 2xy \right )^{2}-2xy\times 3a+\left ( 3a \right ) ^{2}\right )\)                        \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

= \(\left ( 2xy+3a \right )\left ( 4x^{2}y^{2}-6xya+9a^{2} \right )\)

\(∴\) \(8x^{3}y^{3}+27a^{3}\) = \(\left ( 2xy+3a \right )\left ( 4x^{2}y^{2}-6xya+9a^{2} \right )\)

Q 5 . \(64a^{3}-b^{3}\)

SOLUTION :

= \(\left ( 4a \right )^{3}-b^{3}\)

= \(\left ( 4a-b \right )\left ( \left ( 4a \right )^{2} +4a\times b+b^{2}\right )\)                                 \(∵ \left [ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2} +ab+b^{2}\right ) \right ]\)

=\(\left ( 4a-b \right )\left ( 16a ^{2} +4ab+b^{2}\right )\)

\(∴\)  \(64a^{3}-b^{3}\) =\(\left ( 4a-b \right )\left ( 16a ^{2} +4ab+b^{2}\right )\)

Q 6 . \(\frac{x^{3}}{216}-8y^{3}\)

SOLUTION :

= \(\frac{x}{6}^{3}-\left ( 2y \right )^{3}\)

= \(\left ( \frac{x}{6} -2y\right )\left ( \left ( \frac{x}{6} \right )^{2} +\frac{x}{6}\times 2y+\left ( 2y \right )^{2}\right )\)                               \(∵ \left [ x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2} +xy+y^{2}\right ) \right ]\)

=\(\left ( \frac{x}{6} -2y\right )\left ( \frac{x^{2}}{36} +\frac{xy}{3}+ 4y^{2} \right )\)

\(∴\)   \(\frac{x^{3}}{216}-8y^{3}\)  =\(\left ( \frac{x}{6} -2y\right )\left ( \frac{x^{2}}{36} +\frac{xy}{3}+ 4y^{2} \right )\)

Q 7 . \(10x^{4}y-10xy^{4}\)

SOLUTION  :

= \(10xy\left ( x^{3}-y^{3} \right )\)

= \(10xy\left ( x-y\right )\left ( x^{2}+xy+y^{2} \right )\)                          \(∵ \left [ x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2} +xy+y^{2}\right ) \right ]\)

\(∴\) \(10x^{4}y-10xy^{4}\)  = \(10xy\left ( x-y\right )\left ( x^{2}+xy+y^{2} \right )\)

Q 8 . \(54x^{6}y+2x^{3}y^{4}\)

SOLUTION  :

= \(2x^{3}y\left ( 27x^{3}+y^{3} \right )\)

= \(2x^{3}y\left ( \left ( 3x \right )^{3}+y^{3} \right )\)

= \(2x^{3}y\left ( 3x+y \right )\left ( \left ( 3x \right )^{2}-3x\times y+y^{2} \right )\)                  \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(2x^{3}y\left ( 3x+y \right )\left ( 9x ^{2}-3x y+y^{2} \right )\)

\(∴\) \(54x^{6}y+2x^{3}y^{4}\) =\(2x^{3}y\left ( 3x+y \right )\left ( 9x ^{2}-3x y+y^{2} \right )\)

Q 9 . \(32a^{3}+108b^{3}\)

SOLUTION  :

= \(4\left ( 8a^{3}+27b^{3} \right )\)

= \(4\left ( \left ( 2a \right )^{3}+\left ( 3b \right )^{3} \right )\)

= \(4\left [ \left ( 2a+3b \right ) \left ( \left ( 2a \right )^{2}-2a\times 3b+\left ( 3b \right )^{2} \right )\right ]\)                                                \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(4 \left ( 2a+3b \right ) \left ( 4a ^{2}-6a b+ 9b^{2} \right )\)

\(∴\) \(32a^{3}+108b^{3}\) =\(4 \left ( 2a+3b \right ) \left ( 4a ^{2}-6a b+ 9b^{2} \right )\)

Q 10 . \(\left ( a-2b \right )^{3}-512b^{3}\)

SOLUTION  :

= \(\left ( a-2b \right )^{3}-\left ( 8b \right )^{3}\)

= \(\left ( a-2b-8b \right )\left ( \left ( a-2b \right )^{2}+\left ( a-2b \right )8b+\left ( 8b \right ) ^{2}\right )\)                                    \(∵ \left [ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2} +ab+b^{2}\right ) \right ]\)

=\(\left ( a-10b \right )\left ( a^{2}+4b^{2}-4ab +8ab- 16b ^{2}+64b^{2}\right )\)

=\(\left ( a-10b \right )\left ( a^{2}+52b^{2}+4ab \right )\)

\(∴\)  \(\left ( a-2b \right )^{3}-512b^{3}\) =\(\left ( a-10b \right )\left ( a^{2}+52b^{2}+4ab \right )\)

Q 11 . \(\left ( a+b \right )^{3}-8\left ( a-b \right )^{3}\)

SOLUTION  :

= \(\left ( a+b \right )^{3}-\left [ 2\left ( a-b \right ) \right ]^{3}\)

= \(\left ( a+b \right )^{3}-\left [ 2 a-2b \right ]^{3}\)

= \(\left ( a+b-\left ( 2a-2b \right ) \right )\left ( \left ( a+b \right )^{2}+\left ( a+b \right ) \left ( 2a-2b \right )+\left ( 2a-2b \right )^{2}\right )\)                                                                      \(∵ \left [ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2} +ab+b^{2}\right ) \right ]\)

=\(\left ( a+b- 2a+2b \right )\left ( a^{2}+b^{2} +2ab +\left ( a+b \right ) \left ( 2a-2b \right )+\left ( 2a-2b \right )^{2}\right )\)

=\(\left ( a+b- 2a+2b \right )\left ( a^{2}+b^{2} +2ab +2a^{2}-2ab +2ab-2b^{2}+\left ( 2a-2b \right )^{2}\right )\)

=\(\left ( 3b-a \right )\left ( 3a^{2}+2ab-b^{2}+\left ( 2a-2b \right )^{2} \right )\)

=\(\left ( 3b-a \right )\left ( 3a^{2}+2ab-b^{2}+ 4a^{2}+4b^{2}-8ab \right )\)

=\(\left ( 3b-a \right )\left ( 3a^{2}+ 4a^{2}-b^{2}+4b^{2}-8ab+2ab \right )\)

=\(\left ( 3b-a \right )\left ( 7a^{2}+3b^{2}-6ab \right )\)

\(∴\)  \(\left ( a+b \right )^{3}-8\left ( a-b \right )^{3}\) =\(\left ( 3b-a \right )\left ( 7a^{2}+3b^{2}-6ab \right )\)

Q 12 . \(\left ( x+2 \right ) ^{3}+\left ( x-2 \right )^{3}\)

SOLUTION  :

= \(\left ( x+2+x-2 \right )\left ( \left ( x+2 \right ) ^{2}-\left ( x+2 \right )\left ( x-2 \right )+\left ( x-2 \right )^{2}\right )\)                                               \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(2x\left ( x^{2}+4x+4-\left ( x+2 \right )\left ( x-2 \right )+x^{2}-4x+4 \right )\)

=\(2x\left ( 2x^{2}+8-\left ( x^{2}-2^{2} \right )\right )\)                           \(\left [ ∵ \left ( a+b \right )\left ( a-b \right )=a^{2}-b^{2} \right ]\)

=\(2x\left ( 2x^{2}+8-x^{2}+4 \right )\)

=\(2x\left ( x^{2}+12 \right )\)

\(∴\) \(\left ( x+2 \right ) ^{3}+\left ( x-2 \right )^{3}\) =\(2x\left ( x^{2}+12 \right )\)

Q 13 .  \(8x^{2}y^{3}-x^{5}\)

SOLUTION  :

= \(x^{2}\left ( \left ( 2y \right )^{3}-x^{3} \right )\)

= \(x^{2}\left ( 2y -x \right )\left ( \left ( 2y \right )^{2}+2y\times x +x^{2} \right )\)     \(\left [ ∵ x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2}+xy+y^{2} \right ) \right ]\)

= \(x^{2}\left ( 2y -x \right )\left ( 4y ^{2}+2 xy +x^{2} \right )\)

\(∴\) \(8x^{2}y^{3}-x^{5}\) = \(x^{2}\left ( 2y -x \right )\left ( 4y ^{2}+2 xy +x^{2} \right )\)

Q  14 . 1029 – \(3x^{3}\)

SOLUTION  :

= \(3\left ( 343-x^{3} \right )\)

= \(3\left ( \left ( 7 \right )^{3}-x^{3} \right )\)

= \(3\left ( 7-x \right )\left ( 7^{2}+7x+x^{2} \right )\)                                    \(\left [ ∵ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right ) \right ]\)

=\(3\left ( 7-x \right )\left ( 49+7x+x^{2} \right )\)

\(∴\) 1029 – \(3x^{3}\) =\(3\left ( 7-x \right )\left ( 49+7x+x^{2} \right )\)

Q 15 . \(x^{6}+y^{6}\)

SOLUTION  :

= \(\left ( x^{2} \right )^{3}+\left ( y^{2} \right )^{3}\)

= \(\left ( x^{2} +y^{2} \right )\left ( \left ( x ^{2}\right )^{2} -x^{2}y^{2}+\left ( y^{2} \right )^{2}\right )\)

= \(\left ( x^{2}+y^{2} \right )\left ( x ^{4}-x^{2}y^{2}+y^{4} \right )\)                                                \(\left [ ∵ a^{3}+b^{3}=\left ( a+b \right ) \left ( a^{2}-ab+b^{2} \right )\right ]\)

\(∴\) \(x^{6}+y^{6}\) = \(\left ( x^{2}+y^{2} \right )\left ( x ^{4}-x^{2}y^{2}+y^{4} \right )\)

Q 16 . \(x^{3}y^{3}+1\)

SOLUTION  :

= \(\left ( xy \right )^{3}+1^{3}\)

= \(\left ( xy +1 \right )\left ( \left ( xy \right )^{2}+xy+1^{2} \right )\)                                                \(\left [ ∵ x^{3}+y^{3}=\left ( x+y \right )\left ( x^{2}-xy+y^{2} \right ) \right ]\)

=\(\left ( xy +1 \right )\left ( x^{2}y ^{2}-xy+1 \right )\)

\(∴\) \(x^{3}y^{3}+1\) = \(\left ( xy +1 \right )\left ( x^{2}y ^{2}-xy+1 \right )\)

Q 17 . \(x^{4}y^{4}-xy\)

SOLUTION  :

= \(xy\left ( x^{3}y^{3}-1 \right )\)

= \(xy\left ( \left ( xy \right )^{3}-1^{3} \right )\)

= \(xy\left ( xy-1 \right )\left ( \left ( xy \right )^{2}+xy\times 1+1^{2} \right )\)                                             \(∵ \left [ x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2} +xy+y^{2}\right ) \right ]\)

=\(xy\left ( xy-1 \right )\left ( x^{2}y^{2}+xy+1\right )\)

\(∴\) \(x^{4}y^{4}-xy\) = \(xy\left ( xy-1 \right )\left ( x^{2}y^{2}+xy+1\right )\)

Q 18 . \(a^{12}+b^{12}\)

SOLUTION  :

= \(\left ( a^{4} \right )^{3}+\left ( b^{4} \right )^{3}\)

= \(\left ( a^{4} + b^{4} \right )\left ( \left ( a^{4} \right )^{2}-a^{4}\times b^{4}+\left ( b^{4}\right )^{2} \right )\)                                                \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(\left ( a^{4} + b^{4} \right ) \left ( a^{8}-a^{4}b^{4}+ b^{8} \right )\)

\(∴\) \(a^{12}+b^{12}\) = \(\left ( a^{4} + b^{4} \right ) \left ( a^{8}-a^{4}b^{4}+ b^{8} \right )\)

Q 19 . \(x^{3}+6x^{2}+12x+16\)

SOLUTION  :

= \(x^{3}+6x^{2}+12x+8 +8\)

= \(x^{3}+3\times x^{2}\times 2+3\times x\times 2^{2}+2^{3} +8\)

= \(\left ( x+2 \right )^{3} +8\)                                              \(\left [ ∵ a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left ( a+b \right )^{3} \right ]\)

= \(\left ( x+2 \right )^{3}+2^{3}\)

= \(\left ( x+2 +2\right )\left ( \left ( x+2 \right ) ^{2}-2\left ( x+2 \right )+2^{2}\right )\)                            \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(\left ( x+2 +2\right )\left ( x^{2}+4+4x -2x-4+4\right )\)                       \(\left [ ∵ \left ( a+b \right )^{2} =a^{2}+b^{2}+2ab\right ]\)

=\(\left ( x+4\right )\left ( x^{2}+4+2x\right )\)

\(∴\) \(x^{3}+6x^{2}+12x+16\) = \(\left ( x+4\right )\left ( x^{2}+4+2x\right )\)

Q 20 . \(a^{3}+b^{3}+a+b\)

SOLUTION  :

= \(\left ( a^{3}+b^{3} \right )+1\left ( a+b \right )\)

= \(\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right )+1\left ( a+b \right )\)                                         \(\left [ ∵ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2}-ab+b^{2} \right ) \right ]\)

=\(\left ( a+b \right )\left ( a^{2} -ab+b^{2}+1 \right )\)

\(∴\) \(a^{3}+b^{3}+a+b\) = \(\left ( a+b \right )\left ( a^{2} -ab+b^{2}+1 \right )\)

Q 21 . \(a^{3}-\frac{1}{a^{3}}-2a+\frac{2}{a}\)

SOLUTION  :

= \(\left ( a^{3}-\frac{1}{a^{3}} \right )-2\left ( a-\frac{1}{a} \right )\)

= \(\left ( a^{3}-\left ( \frac{1}{a} \right )^{3} \right )-2\left ( a-\frac{1}{a} \right )\)

= \(\left ( a- \frac{1}{a} \right )\left ( a^{2}+a\times \frac{1}{a}+\left ( \frac{1}{a} \right )^{2} \right )-2\left ( a-\frac{1}{a} \right )\)                                             \(\left [ ∵ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right ) \right ]\)

=\(\left ( a- \frac{1}{a} \right )\left ( a^{2}+1+ \frac{1}{a^{2}} \right )-2\left ( a-\frac{1}{a} \right )\)

=\(\left ( a- \frac{1}{a} \right )\left ( a^{2}+1+ \frac{1}{a^{2}}-2 \right )\)

=\(\left ( a- \frac{1}{a} \right )\left ( a^{2}+ \frac{1}{a^{2}}-1 \right )\)

\(∴\) \(a^{3}-\frac{1}{a^{3}}-2a+\frac{2}{a}\) = \(\left ( a- \frac{1}{a} \right )\left ( a^{2}+ \frac{1}{a^{2}}-1 \right )\)

Q 22 . \(a^{3}+3a^{2}b+3ab^{2}+b^{3}-8\)

SOLUTION  :

= \(\left ( a+b \right )^{3}-8\)                                                                \(\left [ ∵ a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left ( a+b \right )^{3} \right ]\)

=\(\left ( a+b \right )^{3}-2^{3}\)

=\(\left ( a+b-2 \right )\left ( \left ( a+b \right )^{2}+\left ( a+b \right )\times 2+2^{2} \right )\)

= (a + b – 2)(a² + 2ab + b² + 2a + 2b + 4)

\(∴\) \(a^{3}+3a^{2}b+3ab^{2}+b^{3}-8\) = (a + b – 2)(a² + 2ab + b² + 2a + 2b + 4)

Q 23 . \(8a^{3}-b^{3}-4ax+2bx\)

SOLUTION  :

= \(\left ( 2a \right )^{3}-b^{3}-2x\left ( 2a-b \right )\)

= \(\left ( 2a-b \right )\left ( \left ( 2a \right )^{2} +2a\times b+b^{2}\right )-2x\left ( 2a-b \right )\)                      \(\left [ ∵ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right ) \right ]\)

=\(\left ( 2a-b \right )\left ( 4a ^{2} +2ab+b^{2}-2x \right )\)

\(∴\) \(8a^{3}-b^{3}-4ax+2bx\)= \(\left ( 2a-b \right )\left ( 4a ^{2} +2ab+b^{2}-2x \right )\)

Q 24 . i . \(\frac{173\times 173\times 173+127\times 127\times 127}{173\times 173-173\times 127+127\times 127}\)

SOLUTION  :

= \(\frac{173^{3}+127^{3}}{173^{2}-173\times 127+127^{2}}\)

= \(\frac{\left ( 173+127 \right )\left ( 173^{2}-173\times 127+127^{2} \right )}{173^{2}-173\times 127+127^{2}}\)                                                \(∵ \left [ a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2} -ab+b^{2}\right ) \right ]\)

=\(\left ( 173+127 \right )\)

=300

Q 24 . ii . \(\frac{1.2\times 1.2\times 1.2-0.2\times 0.2\times 0.2}{1.2\times 1.2+1.2\times 0.2+0.2\times 0.2}\)

SOLUTION  :

= \(\frac{1.2^{3}-0.2^{3}}{1.2^{2}+1.2\times 0.2+0.2^{2}}\)

= \(\frac{\left ( 1.2-0.2 \right )\left ( \left ( 1.2 \right )^{2} +1.2\times 0.2+\left ( 0.2 \right )^{2}\right )}{1.2^{2}+1.2\times 0.2+0.2^{2}}\)                              \(\left [ ∵ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right ) \right ]\)

=\(\left ( 1.2-0.2 \right)\)

=1.0

Q 24 . iii  . \(\frac{155\times 155\times 155-55\times 55\times 55}{155\times 155+155\times 55+55\times 55}\)

SOLUTION  :

= \(\frac{155^{3}-55^{3}}{155^{2}+155\times 55+55^{2}}\)

= \(\frac{\left (155-55 \right )\left ( 155^{2}+155\times 55+55^{2}\right )}{155^{2}+155\times 55+55^{2}}\)                                     \(\left [ ∵ a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right ) \right ]\)

= (155 – 55)

= 100