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  • Factorize (i) a^{3}-8b^{3}-64c^{3}-24abc (ii) 2\sqrt{2}a^{3}+8b^{3}-27c^{3}+18\sqrt{2}abc

Factorize

(i) a^{3}-8b^{3}-64c^{3}-24abc

(ii) 2\sqrt{2}a^{3}+8b^{3}-27c^{3}+18\sqrt{2}abc

Answers (1)

(i) (a-2b-4c)(a^{2}+4b^{2}+16c^{2}+2ab-8bc+4ac)

Solution:

Given: a^{3}-8b^{3}-64c^{3}-24abc

This can be written as:

a^{3}+(-2b)^{3}+(-4c)^{3}-3(a)(-2b)(-4c)

We know that

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

Using this identity we get

a^{3}+(-2b)^{3}+(-4c)^{3}-3(a)(-2b)(-4c)

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

=(a-2b-4c)(a^{2}+4b^{2}+16c^{2}+2ab-8bc+4ac)

         

(i) (\sqrt{2}a+2b-3c)(2a^{2}+4b^{2}+9c^{2}-2\sqrt{2}ab+6bc+3\sqrt{2}ca)

Solution:

Given: 2\sqrt{2}a^{3}+8b^{3}-27c^{3}+18\sqrt{2}abc

This can be written as:

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

We know that

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

Using this identity we get

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

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

= (\sqrt{2}a+2b-3c)(2a^{2}+4b^{2}+9c^{2}-2\sqrt{2}ab+6bc+3\sqrt{2}ca)

Posted by

infoexpert24

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