Question

Solve pls no spams pls

Answer 1

Step-by-step explanation:

★ Concept :-

Here the concept of algebraic identities has been used. We see that we are given an equation to factorise in simplest form. So firstly we can expand the equation to it's original form. The we shall start taking the common terms. After that we shall apply different algebraic identities to make groups. Then we shall follow the procedure of grouping and thus find our answer.

Let's do it !!

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★ Solution :-

Given,

» (x + y)² (x - y)² - 18(x² + y²) + 81

By taking the square in common for first term, this can be written as,

>> [(x + y)(x - y)]² - 18(x² + y²) + 81

>> [x² - xy + xy - y²]² - 18(x² + y²) + 81

By cancelling the unlike terms, we get

>> [x² - y²]² - 18(x² + y²) + 81

• • Identity 1 : (a - b)² = a² + b² - 2ab

Here a = x²

Here b = y²

By applying these values, we get

>> [(x²)² + (y²)² - 2(x²)(y²)] - 18(x² + y²) + 81

>> [x⁴ + y⁴ - 2x²y²] - 18(x² + y²) + 81

Now opening the brackets of all the terms, we get

>> x⁴ + y⁴ - 2x²y² - 18x² - 18y² + 81

Now here we need to make squares of every term so that their initial value doesn't change. So,

>> (x²)² + (y²)² - 2x²y² - 2[9x² + 9y²] + (9)²

>> (x²)² + (y²)² - 2x²y² - 2[(3x)² + (3y)²] + (9)²

Here we see that, first three terms forms like the Identity 1 as : (a - b)² = a² + b² - 2ab

• Here a = (x²)
• Here b = (y²)

Now taking groups, we get

>> (x² - y²)² - 2[9x² + 9y²] + 9²

Now again we see that this equation is similar to Identity 1.

• Here a = (x² - y²)²
• Here b = (9)

Now on using Identity 1 by applying these values, we get

>> [(x² - y²) - 9]²

This is the required answer.

$\;\underline{\boxed{\tt{Required\;\: Factor\;=\;\bf{\purple{[(x^{2}\:-\:y^{2})\:-\:9]^{2}}}}}}$

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★ More to know :-

• (a + b)² = a² + b² + 2ab
• a² - b² = (a + b)(a - b)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac
• (a + b)³ = a³ + b³ + 3ab(a + b)
• • (a - b)³ = a³ - b³ - 3a²b + 3ab²