Name: Using Algebraic Expressions in the Real World
Uploaded: 2019-07-19
Duration: 2 min 41 s
Description: Algebraic expressions and the terms within them are not just for math class – there are plenty of real-world applications. You may even find a career where y...

Question

Factorise

x^{4} (y - z) + y^{4} (z - x) + z^{4} (x - y)

.

Accepted Answer

We are given to factorise $x^{4} (y - z) + y^{4} (z - x) + z^{4} (x - y)$ .

So, we get, $x^{4} (y - z) + y^{4} (z - x) + z^{4} (x - y)$

$= x^{4} (y - z) + y^{4} z - x y^{4} + z^{4} x - y z^{4}$

$= x^{4} (y - z) - x y^{4} + z^{4} x + y^{4} z - y z^{4}$

$= x^{4} (y - z) - x (y^{4} - z^{4}) + y z (y^{3} - z^{3})$

$= x^{4} (y - z) - x (y^{2} - z^{2}) (y^{2} + z^{2}) + y z (y - z) (y^{2} + y z + z^{2})$ [Use $a^{2} - b^{2} = (a + b) (a - b)$ and $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ ]

$= x^{4} (y - z) - x (y - z) (y + z) (y^{2} + z^{2}) + y z (y - z) (y^{2} + y z + z^{2})$ [Use $a^{2} - b^{2} = (a + b) (a - b)$ ]

$= (y - z) [x^{4} - x (y + z) (y^{2} + z^{2}) + y z (y^{2} + y z + z^{2})]$

$= (y - z) [x^{4} - x (y^{3} + y z^{2} + y^{2} z + z^{3}) + y^{3} z + y^{2} z^{2} + y z^{3}]$

$= (y - z) [x^{4} - x y^{3} - x y z^{2} - x y^{2} z - x z^{3} + y^{3} z + y^{2} z^{2} + y z^{3}]$

$= (y - z) [x^{4} - x y^{3} - x y z^{2} + y^{2} z^{2} - x y^{2} z + y^{3} z - x z^{3} + y z^{3}]$

$= (y - z) [x (x^{3} - y^{3}) - y z^{2} (x - y) - y^{2} z (x - y) - z^{3} (x - y)]$

$= (y - z) [x (x - y) (x^{2} + x y + y^{2}) - y z^{2} (x - y) - y^{2} z (x - y) - z^{3} (x - y)]$ [Use $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ ]

$= (y - z) [(x - y) \{x (x^{2} + x y + y^{2}) - y z^{2} - y^{2} z - z^{3}\}]$

$= (x - y) (y - z) \{x (x^{2} + x y + y^{2}) - y z^{2} - y^{2} z - z^{3}\}$

$= (x - y) (y - z) (x^{3} + x^{2} y + x y^{2} - y z^{2} - y^{2} z - z^{3})$

$= (x - y) (y - z) (x^{3} - z^{3} + x^{2} y - y z^{2} + x y^{2} - y^{2} z)$

$= (x - y) (y - z) [(x - z) (x^{2} + z x + z^{2}) + y (x^{2} - z^{2}) + y^{2} (x - z)]$ [Use $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ ]

$= (x - y) (y - z) [(x - z) (x^{2} + z x + z^{2}) + y (x + z) (x - z) + y^{2} (x - z)]$ [Use $a^{2} - b^{2} = (a + b) (a - b)$ ]

$= (x - y) (y - z) [(x - z) \{x^{2} + z x + z^{2} + y (x + z) + y^{2}\}]$

$= (x - y) (y - z) (x - z) (x^{2} + z x + z^{2} + x y + y z + y^{2})$

$= - (x - y) (y - z) (z - x) (x^{2} + y^{2} + z^{2} + x y + y z + z x)$ .

Therefore, the required factorisation is $- (x - y) (y - z) (z - x) (x^{2} + y^{2} + z^{2} + x y + y z + z x)$ .

Factorise $x^{4} (y - z) + y^{4} (z - x) + z^{4} (x - y)$ .

Important Questions on Factorisation

Learn and Prepare for any exam you want !

Factorise x4(y-z)+y4(z-x)+z4(x-y).

Important Questions on Factorisation

Learn and Prepare for any exam you want !

Factorise $x^{4} (y - z) + y^{4} (z - x) + z^{4} (x - y)$ .