Question

factorize (1-a^2)(1-b^2)+4ab

Answer 1

Concept

The word "factor" is used to describe a number as the result of any two other numbers. Any mathematical object, including numbers, polynomials, and algebraic expressions, can have factors by using the factorization method. Finding the factors of a given algebraic expression is referred to as factorization of an algebraic expression.

Given

the expression is (1 ₋ a²)(1 ₋ b²) ₊ 4ab

Find

we need to factorize the given expression.

Solution

given,

(1 ₋ a²)(1 ₋ b²) ₊ 4ab

multiply the above terms.

= 1(1 ₋ b²) ₋ a²(1 ₋ b²) ₊ 4ab

= 1 ₋ b² ₋ a² ₊ a²b² ₊ 4ab

now split 4ab into two terms

= 1 ₋ b² ₋ a² ₊ a²b² ₊ 2ab ₊ 2ab

combine the terms

= ( 1 ₊ a²b² ₊2ab) ₋ (a² ₊ b² ₋ 2ab)

the above expression represents the following form:

( a ₊ b)² = a² ₊ b² ₊ 2ab

( a ₋ b)² = a² ₊ b² ₋ 2ab

hence they can be written as

(1 ₊ ab)² ₋ (a ₋ b)²

now the above expression is in the form of a² ₋ b² = (a₊b)(a₋b)

therefore, (1 ₊ ab ₊ a ₋ b) (1 ₊ ab ₋ (a ₋b))

= (1 ₊ ab ₊ a ₋ b) (1 ₊ ab ₋ a ₊ b)

Consequently, when factorizing, we obtain the desired result.

#SPJ2

Answer 2

Answer:

The required factors are (1 + ab + a - b) and (1 + ab - a + b).

Step-by-step explanation:

Factorisation or factoring: It is defined as the breaking of or decomposition of an entity (a number or a polynomial) into a product of another entity or factors which when multiplied together gives the original entity.

Step 1 of 3

Consider the given expression as follows:

(1 - a²)(1 - b²) + 4ab

Multiply the terms of the brackets as follows:

⇒ 1(1 - b²) - a²(1 - b²) + 4ab

⇒ 1 - b² - a² - (-a²b²) + 4ab

⇒ 1 - b² - a² + a²b² + 4ab

Now, rewrite the expression as follows:

⇒ 1 - b² - a² + a²b² + 2ab + 2ab

Rearrange the terms as follows:

⇒ 1 + a²b² + 2ab - b² - a² + 2ab

⇒ (1 + a²b² + 2ab) - (a² + b² - 2ab) . . . . . (1)

Step 2 of 3

Now, consider the expression of the first bracket.

(1 + a²b² + 2ab)

(1² + (ab)² + 2 × 1 × ab)

(1 + ab)² (Using the identity, $(a+b)^2=a^2+b^2+2ab$)

Similarly,

Simplify the expression of the second bracket.

(a² + b² - 2ab)

(a² + b² - 2 × a × b)

(a - b)² (Using the identity, $(a-b)^2=a^2+b^2-2ab$)

Step 3 of 3

Substitute the values of (1 + a²b² + 2ab) and (a² + b² - 2ab) in the (1), we get

⇒ (1 + ab)² - (a - b)²

Notice that the expression is written in the form of $a^2-b^2$.

Using the identity, $a^2-b^2=(a+b)(a-b)$, we get

⇒ (1 + ab + a - b)(1 + ab - a + b)

Therefore, the required factors are (1 + ab + a - b) and (1 + ab - a + b).

#SPJ2