Question

factorise:25x^2y^2-20xy^2z+4y^2z^2​

Answer 1

Answer:

y^2(5x-2z)^2

Step-by-step explanation:

Given an algebraic expression such that,

$25 {x}^{2} {y}^{2} - 20x {y}^{2} z + 4 {y}^{2} {z}^{2}$

To factorise this.

Simplifying further, we will get,

$= {5}^{2} {x}^{2} {y}^{2} - 20x {y}^{2} z + {2}^{2} {y}^{2} {z}^{2} \\ \\ = {(5xy)}^{2} - 2(5xy)(2yz) + {(2yz)}^{2}$

It's in the form of

• ${a}^{2} - 2ab + {b}^{2}$

But, we know that,

${a}^{2} - 2ab + {b}^{2} = {(a - b)}^{2}$

Therefore, we will get,

$= {(5xy - 2yz)}^{2} \\ \\ = {(y (5x - 2z))}^{2} \\ \\ = {y}^{2} {(5x - 2z)}^{2}$

Hence, the required answer is y^2(5x-2z)^2

Answer 2

Given,

We need to factorise:-

$25 {x}^{2} {y}^{2} - 20x {y}^{2} z + 4 {y}^{2} {z}^{2}$

SOLUTION:-

We can write 25 as -

$25 = {(5)}^{2}$

We can write 4 as -

$4 = {(2)}^{2}$

We know that 25 and 4 are perfect squares of 5 and 2 respectively

NOW,

We can write

${x}^{2} {y}^{2} \: \: as \: \: {(xy)}^{2}$

AND,

${y}^{2} {z}^{2} \: \: as \: \: {(yz)}^{2}$

The mathematical statement 25x^2y^2-20xy^2z+4y^2z^2 can be written in the form -

$\implies25 {x}^{2} {y}^{2} - 20x {y}^{2} z + 4 {y}^{2} {z}^{2} \\ \implies {(5xy)}^{2} - (2 \times 5xy \times 2yz) + {(2yz)}^{2}$

We know that,

$\implies {(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2}$

In the given question,

a = 5xy and b = 2yz

So,

$\implies {(5xy)}^{2} - (2 \times 5xy \times 2yz) + {(2yz)}^{2} \\ \implies {(5xy - 2yz)}^{2}$

Taking y as common from the terms 5xy and 2yz , we get,

$\implies {[y(5x - 2z)]}^{2} \\ \implies {y}^{2} {(5x - 2z)}^{2} = ANSWER$

MORE MATHEMATICAL EXPANSIONS:-

• ${(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}$
• ${(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2}$
• ${a}^{2} - {b}^{2} = (a + b)(a - b)$
• ${(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)$
• ${(a - b)}^{3} = {a}^{3} - {b}^{3} - 3ab(a - b)$
• ${(a + b + c)}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca)$
• ${a}^{3} + {b}^{3} = (a + b)( {a}^{2} - ab + {b}^{2} )$
• ${a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2} )$
• If (a+b+c=0) , then , a^3 + b^3 + c^3 = 3abc.