RD Sharma Solutions Class 9 Factorization Of Algebraic Expressions Exercise 5.3

RD Sharma Solutions Class 9 Chapter 5 Exercise 5.3

RD Sharma Class 9 Solutions Chapter 5 Ex 5.3 Download

Q 1 . \(64a^{3}+125b^{3}+240a^{2}b+300ab^{2}\)

SOLUTION  :

= \(\left ( 4a \right )^{3}+\left ( 5b \right )^{3}+3\left ( 4a \right )^{2}\left ( 5b \right )+3\left ( 4a \right )\left ( 5b \right )^{2}\)                                       \(\left [ ∵ a^{3}+b^{3}+3a^{2}b+3ab^{2}=\left ( a +b\right )^{3} \right ]\)

= \(\left ( 4a+5b \right )^{3}\)

= \(\left ( 4a+5b \right )\left ( 4a+5b \right )\left ( 4a+5b \right )\)

\(∴\) \(64a^{3}+125b^{3}+240a^{2}b+300ab^{2}\)  = \(\left ( 4a+5b \right )\left ( 4a+5b \right )\left ( 4a+5b \right )\)

 

Q 2 . \(125x^{3}-27y^{3}-225x^{2}y+135xy^{2}\)

SOLUTION  :

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

=\(\left ( 5x-3y \right )^{3}\)

=\(\left ( 5x-3y \right )\left ( 5x-3y \right )\left ( 5x-3y \right )\)

\(∴\) \(125x^{3}-27y^{3}-225x^{2}y+135xy^{2}\) = \(\left ( 5x-3y \right )\left ( 5x-3y \right )\left ( 5x-3y \right )\)

 

Q 3 . \(\frac{8}{27}x^{3}+1+\frac{4}{3}x^{2}+2x\)

SOLUTION  :

= \(\left ( \frac{2}{3}x^{3} \right )^{3}+1^{3}+3\times \left ( \frac{2}{3}x \right )^{2}\times 1+3\left ( 1 \right )^{2}\times \left ( \frac{2}{3} x\right )\)

=\(\left ( \frac{2}{3}x+1 \right )^{3}\)                                                \(\left [ ∵ x^{3}+b^{3}+3x^{2}b+3xb^{2}=\left ( x +b\right )^{3} \right ]\)

=\(\left ( \frac{2}{3}x+1 \right )\left ( \frac{2}{3}x+1 \right )\left ( \frac{2}{3}x+1 \right )\)

\(∴\) \(\frac{8}{27}x^{3}+1+\frac{4}{3}x^{2}+2x\) = \(\left ( \frac{2}{3}x+1 \right )\left ( \frac{2}{3}x+1 \right )\left ( \frac{2}{3}x+1 \right )\)

 

Q 4 . \(8x^{3}+27y^{3}+36x^{2}y+54xy^{2}\)

SOLUTION  :

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

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

a³ + b³ + 3a²b + 3ab² = (a + b) ³]

=\(\left ( 2x+3y \right )\left ( 2x+3y \right )\left ( 2x+3y \right )\)

\(∴\) \(8x^{3}+27y^{3}+36x^{2}y+54xy^{2}\) = \(\left ( 2x+3y \right )\left ( 2x+3y \right )\left ( 2x+3y \right )\)

 

Q 5 . \(a^{3}-3a^{2}b+3ab^{2}-b^{3}+8\)

SOLUTION  :

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

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

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

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

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

 

Q 6 . \(x^{3}+8y^{3}+6x^{2}y+12xy^{2}\)

SOLUTION  :

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

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

=\(\left ( x+2y \right )\left ( x+2y \right )\left ( x+2y \right )\)

\(∴\) \(x^{3}+8y^{3}+6x^{2}y+12xy^{2}\) = \(\left ( x+2y \right )\left ( x+2y \right )\left ( x+2y \right )\)

 

Q 7 . \(8x^{3}+y^{3}+12x^{2}y+6xy^{2}\)

SOLUTION  :

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

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

=\(\left ( 2x +y \right )\left ( 2x +y \right )\left ( 2x +y \right )\)

\(∴\) \(8x^{3}+y^{3}+12x^{2}y+6xy^{2}\) = \(\left ( 2x +y \right )\left ( 2x +y \right )\left ( 2x +y \right )\)

 

Q 8 . \(8a^{3}+27b^{3}+36a^{2}b+54ab^{2}\)

SOLUTION  :

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

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

=\(\left ( 2a + 3b \right )\left ( 2a + 3b \right )\left ( 2a + 3b \right )\)

\(∴\) \(8a^{3}+27b^{3}+36a^{2}b+54ab^{2}\) =\(\left ( 2a + 3b \right )\left ( 2a + 3b \right )\left ( 2a + 3b \right )\)

 

Q 9 . \(8a^{3}-27b^{3}-36a^{2}b+54ab^{2}\)

SOLUTION  :

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

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

=\(\left ( 2a – 3b \right )\left ( 2a – 3b \right )\left ( 2a – 3b \right )\)

\(∴\) \(8a^{3}-27b^{3}-36a^{2}b+54ab^{2}\) =\(\left ( 2a – 3b \right )\left ( 2a – 3b \right )\left ( 2a – 3b \right )\)

 

Q 10 . \(x^{3}-12x\left ( x-4\right )-64\)

SOLUTION  :

= \(x^{3}-12x^{2}+48x-64\)

= \(x^{3}-3\times x^{2}\times 4+3\times 4^{2}\times x-4^{3}\)

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

=\(\left ( x-4 \right )\left ( x-4 \right )\left ( x-4 \right )\)

\(∴\) \(x^{3}-12x\left ( x-4\right )-64\) =\(\left ( x-4 \right )\left ( x-4 \right )\left ( x-4 \right )\)

 

Q 11 . \(a^{3}x^{3}-3a^{2}bx^{2}+3ab^{2}x-b^{3}\)

SOLUTION  :

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

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

=\(\left ( ax-b \right )\left ( ax-b \right )\left ( ax-b \right )\)

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