Problems on Factorization using a^2 – b^2 | Examples on Factorizing using Identity a^2 – b^2 = (a+b)(a-b)

In this article of ours, you will find various problems on finding the factorization using a² – b². Learn how to approach when you are given an algebraic expression using the basic identity a² – b² = (a+b)(a-b). Solve the problems available here and learn different models of questions framed on the topic. Master the concept and learn factor algebraic expressions much easily by employing these simple identities. Try to answer the questions on your own and then cross-check with our solutions.

Do Refer:

• Problems on Factorization by Grouping of Terms
• Problems on Factorization of Expressions of the Form a2 – b2

Questions on Factorizing using a² – b² Identity

Example 1.
Factorize: 16a² – b² + 4a + b
Solution:
Given expression = 16a² – b² + 4a + b
= (16a² – b²) + 4a + b
= {(4a)² – b²} + 4a + b
= (4a + b)(4a – b) + 1(4a + b)
= (4a + b)(4a – b + 1)

Example 2.
Factorize: x³ – 5x² – x + 5
Solution:
Given expression = x³ – 5x² – x + 5
= (x³ – 5x²) – x + 5
= x²(x – 5) – 1(x – 5)
= (x – 5)(x² – 1)
= (x – 5)(x² – 1²)
= (x – 5)(x + 1)(x – 1)

Example 3.
Factorize: 5x² – y²+ 2x – 2y – 4xy
Solution:
Given expression = 5x² – y²+ 2x – 2y – 4xy
= x² – y² + 2x – 2y + 4x² – 4xy
= (x + y)(x – y) + 2(x – y) + 4x(x – y)
= (x – y)(x + y + 2 + 4x)
= (x – y)(5x + y + 2)

Example 4.
Factorize: a⁴+ a²b² + b⁴
Solution:
Given expression = a⁴ +a²b² + b⁴
= a⁴+2a²b² -a²b²+ b⁴
= (a²)² + 2 ∙ a² ∙ b² + (b²)² – a²b²
= (a² + b²)² – (ab)²
= (a² + b² + ab)( a² + b² – ab)

Example 5.
Factorize x² + xy – 4y – 16
Solution:
Given Expression = x² + xy – 4y – 16
= (x² – 16) + xy – 4y
= (x² – 4²) + y(x – 4)
= (x + 4)(x – 4) + y(x – 4)
= (x – 4)(x +4 + y)
= (x – 4)(x + y + 4)

Example 6.
Factorize x(x – 6) – y(y – 6)
Solution:
Given Expression = x(x – 6) – y(y – 6)
=x²-6x-y²+6y
= (x² – y²) – 6x + 6y
= x² – 6x – y² + 6y
= (x + y)(x – y) – 6(x – y)
= (x – y)(x + y – 6)

Example 7.
Factorize a⁴  + 49?
Solution:
Given Expression = a⁴  + 49
=(a²)² + 7²
= (a²)² + 2 ∙ a² ∙ 7 + 7² – 2 ∙ a² ∙ 7
= (a² + 7)² – 14a²
= (a² + 7)² – (√14a)²
= (a²+ 7 + √14a)(a² + 7 – √14a)
=(a²+ 7 + √14a)(a² + 7 – √14a)

Example 8.
Express x² – 5x + 6?
Solution:
Given Expression = x² – 5x + 6
= x² -3x-2x+6
= x(x-3)-2(x-3)
=(x-2)(x-3)