Class 9th Maths NCERT chapter 02 Polinomials Exercise 2.4 Q.N 11 to 15

*Exercise 2.4: Factorization Using Identities*
In this exercise, you will focus on factorizing polynomials using algebraic identities. Understanding and applying these identities can simplify complex polynomial expressions and make factorization more straightforward.
*Objectives:*
1. To recognize and use standard algebraic identities for factorization.
2. To simplify polynomial expressions using these identities.
3. To solve polynomial equations by applying factorization techniques.
*Standard Algebraic Identities:*
1. *Square of a Binomial:*
- \((a + b)^2 = a^2 + 2ab + b^2\)
- \((a - b)^2 = a^2 - 2ab - b^2\)
2. *Difference of Squares:*
- \(a^2 - b^2 = (a - b)(a + b)\)
3. *Perfect Square Trinomials:*
- \(a^2 + 2ab + b^2 = (a + b)^2\)
- \(a^2 - 2ab + b^2 = (a - b)^2\)
4. *Sum of Cubes:*
- \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
5. *Difference of Cubes:*
- \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
*Instructions:*
1. *Identify the Form:* Begin by identifying which algebraic identity the given polynomial matches.
2. *Apply the Identity:* Use the appropriate algebraic identity to factorize the polynomial.
3. *Simplify the Expression:* Write the factorized form and verify by expanding it to ensure it matches the original polynomial.
*Example Problems:*
1. Factorize \(x^2 - 25\).
- Identify as a difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).
- Solution: \(x^2 - 25 = (x - 5)(x + 5)\).
2. Factorize \(9x^2 + 6x + 1\).
- Identify as a perfect square trinomial: \(a^2 + 2ab + b^2 = (a + b)^2\).
- Solution: \(9x^2 + 6x + 1 = (3x + 1)^2\).
3. Factorize \(8a^3 - 27b^3\).
- Identify as a difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
- Solution: \(8a^3 - 27b^3 = (2a - 3b)(4a^2 + 6ab + 9b^2)\).
*Hints:*
- Look for patterns that match the standard algebraic identities.
- Practice recognizing which identity applies to a given polynomial expression.
*Use these identities to efficiently factorize polynomials and solve related problems, enhancing your algebraic skills.*
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