Question

Factorise the following using suitable identity (a/b)^3 + (b/c)^3 +(c/a)^3 -3

Guys please help ​

Answer 1

$\red{\bigstar}$Question

Factorize the following using suitable identities :

$\blue{\bullet}\:\:\:\sf{(\frac{a}{b})^3+(\frac{b}{c})^3+(\frac{c}{a})^3-3}$

$\red{\bigstar}$Solution

Given expression is

$\blue{\sf{\implies (\frac{a}{b})^3+(\frac{b}{c})^3+(\frac{c}{a})^3-3 }}$

It can be also written as

$\blue{\sf{\implies (\frac{a}{b})^3+(\frac{b}{c})^3+(\frac{c}{a})^3-3(1)}}$

Now, Since

$\sf{(\frac{a}{b})\times(\frac{b}{c})\times(\frac{c}{a})=1}$

Therefore, expression can be written as

$\blue{\sf{\implies (\frac{a}{b})^3+(\frac{b}{c})^3+(\frac{c}{a})^3-3(\frac{a}{b})(\frac{b}{c})(\frac{c}{a})}}$

Now using algebraic identity

$\boxed{\begin{minipage}{7cm} x^3+y^3+z^3-3xyz=\\(x+y+z)(x^2+y^2+z^2-(xy+yz+xz) )\end{minipage}}$

we will get

$\blue{\sf{\implies \; (\frac{a}{b}+\frac{b}{c}+\frac{c}{a})(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}-(\frac{a}{b}\times\frac{b}{c}+\frac{b}{c}\times\frac{c}{a}+\frac{c}{a}\times\frac{a}{b}))}}$

On simplifying we will get

$\blue{\sf{\implies \; (\frac{a}{b}+\frac{b}{c}+\frac{c}{a})(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}-(\frac{a}{c}+\frac{b}{a}+\frac{c}{b}))}}$

$\pink{\star}$Factorised .

$\red{\bigstar}$Know more algebraic

identites

$\boxed{\begin{minipage}{7cm}\rule{200}1 \bullet(x+y)^2=x^2+y^2+2xy \\\rule{200}1\\ \bullet(x-y)^2=x^2+y^2-2xy \\\rule{200}1\\\bullet x^2-y^2=(x+y)(x-y)\\\rule{200}1\\\bullet (x+a)(x+b)=x^2+x(a+b)+ab\\\rule{200}1 \\\bullet (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+xz)\\\rule{200}1\\\bullet (x+y)^3=x^3+y^3+3xy(x+y)\\\rule{200}1\\\bullet (x-y)^3=x^3-y^3-3xy(x-y)\\\rule{200}1 \end{minipage}}$

Answer 2

Answer:

Please join And play with me ludo king.