Chapter 4: 9th Class Math Notes

Chapter 4: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about factorization and algebraic manipulation.

This chapter requires knowledge of real numbers, including operations with algebraic fractions, roots of quadratic expressions, and some techniques to solve the variables.

Chapter 4: 9th Class Math Notes

Class 9 Math Notes – Chapter 4 Summary

This summary covers key concepts and solution methods from Chapter 4 of your 9th class mathematics notes:

Factorization by Identifying Common Factors

• Objective: Factorize expressions by extracting the highest common factor (HCF).
• Examples:
  • 6x+12=6(x+2)
  • 15y²+20y=5y(3y+4)
  • 4a²b+8ab²=4ab(a+2b)

2. Factorization of Quadratic Expressions

• Objective: Factorize quadratic expressions of the form ax²+bx+c.
• Methods:
  • Splitting the middle term: Break bx into two terms whose coefficients multiply to a×c.
  • Examples:
    • x²+4x+3=(x+1)(x+3)
    • x²−6x+8=(x−4)(x−2)
    • x²−x−56=(x−8)(x+7)

3. Factorization of Higher-Degree Polynomials

• Objective: Factorize polynomials of degree 3 or higher.
• Examples:
  • 8x³+12x²+6x+1=(2x+1)³
  • x³+64y³=(x+4y)(x²−4xy+16y²)
  • x⁶−27=(x²−3)(x⁴+3x²+9)

Special Factorization Techniques

• Objective: Factorize expressions using special formulas.
• Formulas:
  • a²−b²=(a−b)(a+b)
  • a³+b³=(a+b)(a²−ab+b²)
  • a³−b³=(a−b)(a²+ab+b²)
• Examples:
  • 4x⁴+81y⁴=(2x²+9y²−6xy)(2x²+9y²+6xy)
  • x⁴+4x²+16=(x²+4−2x)(x²+4+2x)

HCF and LCM of Polynomials

• Objective: Find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of polynomials.
• Methods:
  • Prime Factorization: Break down each polynomial into its prime factors.
  • Division Method: Use long division to find HCF.
• Examples:
  • HCF of 21x²y and 35xy² is 7xy.
  • LCM of 4x²−9y² and 2x²−3xy is 4x²(x+3)(x+2).

Square Root of Polynomials

• Objective: Find the square root of polynomials using factorization or division.
• Examples:
  • √(x²−8x+16)=±(x−4)
  • √(9x²+12x+4)=±(3x+2)
  • √(16x⁴+8x²+1)=±(4x²+1)

Applications of Factorization

• Objective: Solve real-world problems using factorization.
• Examples:
  • Investment Return: R(x)=−x²+6x−8 factors to (−x+2)(x−4). Zero return occurs at x=2 and x=4.
  • Profit Analysis: P(x)=x³−15x²+75x−125 factors to (x−5)³. Break-even point is at x=5.

Review and Practice

• Objective: Test understanding with multiple-choice questions and factorization problems.
• Examples:
  • Factorize 4x³+18x²−12x=2x(2x²+9x−6).
  • Find LCM and HCF of x³+3x²−4x and x²−x−6.

9th Class Math Notes Full Book