Question

The LCM and HCF of two numbers p(x) and q(x) are 27x^3(x+a)(x^3-a^3) and x^2(x-a) respectively.If p(x)=3x^2(x^2-a^2),find q(x).​

Answer 1

Answer:

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Answer 2

q(x) = 9x³(x³ - a³)

Given :

• The LCM and HCF of two numbers p(x) and q(x) are 27x³(x + a)(x³ - a³) and x²(x -a) respectively

• p(x) = 3x²(x² - a²)

To find :

The value of q(x)

Solution :

Step 1 of 2 :

Factorise LCM and p(x)

Here it is given that the LCM and HCF of two numbers p(x) and q(x) are 27x³(x + a)(x³ - a³) and x²(x -a) respectively

LCM

= 27x³(x + a)(x³ - a³)

= 27x³(x + a)(x - a)(x² + xa + a²)

Also , p(x)

= 3x²(x² - a²)

= 3x²(x + a)(x - a)

Step 2 of 2 :

Find the value of q(x)

We know that ,

p(x).q(x) = LCM × HCF

$\displaystyle \sf{ \implies }3 {x}^{2}( {x}^{2} - {a}^{2}).q(x) = 27 {x}^{3} (x + a)( {x}^{3} - {a}^{3} ). {x}^{2} (x - a)$

$\displaystyle \sf{ \implies }q(x) = \frac{27 {x}^{3} (x + a)( {x}^{3} - {a}^{3} ). {x}^{2} (x - a)}{3 {x}^{2}( {x}^{2} - {a}^{2})}$

$\displaystyle \sf{ \implies }q(x) = \frac{27 {x}^{5} (x + a)(x - a)( {x}^{2} + ax + {a}^{2} )(x - a)}{3 {x}^{2}( x + a)(x - a)}$

$\displaystyle \sf{ \implies }q(x) = \frac{9 {x}^{3} (x + a)(x - a)( {x}^{2} + ax + {a}^{2} )(x - a)}{( x + a)(x - a)}$

$\displaystyle \sf{ \implies }q(x) = 9 {x}^{3} (x - a)( {x}^{2} + ax + {a}^{2} )$

$\displaystyle \sf{ \implies }q(x) = 9 {x}^{3}( {x}^{3} - {a}^{3} )$

∴ q(x) = 9x³(x³ - a³)

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