Notes: Exercise 2.4

Introduction

• Beginning of Exercise 2.4.
• Possibility of dividing a long problem into two parts.

First Question

• Extract product using appropriate identity.
• First identity:
  • ( (x + a)(x + b) = x^2 + (a + b)x + ab )

Problems:

1. ( (x + 4)(x + 10) )

   • ( x^2 + (4 + 10)x + (4 * 10) )
   • Answer: ( x^2 + 14x + 40 )

2. ( (x + 8)(x - 10) )

   • Using identity:
   • ( x^2 + (8 - 10)x + (8 * -10) )
   • Answer: ( x^2 - 2x - 80 )

3. ( (3x + 4)(3x - 5) )

   • Using identity:
   • ( (3x)^2 + (4 - 5)(3x) + (4 * -5) )
   • Answer: ( 9x^2 - 3x - 20 )

4. ( (y^2 + 3/2)(y^2 - 3/2) )

   • Using identity:
   • ( (y^2)^2 - (3/2)^2 )
   • Answer: ( y^4 - (9/4) )

5. ( (3 - 2x)(3 + 2x) )

   • Using identity:
   • ( (3)^2 - (2x)^2 )
   • Answer: ( 9 - 4x^2 )*

Second Question

• Calculate the product without direct multiplication.

Problems:

1. ( 103 * 107 )

   • Using appropriate identity:
   • ( (100 + 3)(100 + 7) = 10000 + 10 + 21 = 1121 )

2. ( 95 * 96 )

   • ( (100 - 5)(100 - 4) )
   • Answer: ( 9120 )

3. ( 104 * 96 )

   • ( (100 + 4)(100 - 4) )
   • Answer: ( 9984 )*

Third Question

• Learn how to factorize the following.
• Identity:
  • ( a^2 - b^2 = (a + b)(a - b) )

Problems:

1. ( x^2 + xy + y^2 )
   • Identity:
   • ( (x + y)^2 - xy )
   • Answer: ( (x + y + z)(x - y)(x - z) )

Fourth Question

• Expand each using appropriate identity.

Problems:

1. ( (a + b + c)^2 )
   • Identity:
   • ( a^2 + b^2 + c^2 + 2ab + 2bc + 2ca )

Other Questions

• Using identity in other questions.

Conclusion

• Importance of identity and recognizing it.
• Participate in more exercises.

Suggestions

• Practice.
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