Lecture on Polynomials

Introduction

• Key Concepts Covered: Polynomials
• Solving important questions and tips/tricks

Zeros of a Polynomial

• Definition: Zeros of a polynomial are the values of X which make the polynomial zero.
• Example Polynomial: (P(x) = x^2 - 5x + 6)
  • Set the polynomial to zero: (x^2 - 5x + 6 = 0)
  • Factorize: (x^2 - 3x - 2x + 6 = 0)
  • Results: ((x - 3)(x - 2) = 0)
  • Zeros: (x = 2) or (x = 3)

Terms: Zeros and Roots

• Zeros and Roots of a polynomial are the same.
• To find zeros: Set polynomial (P(x)) to zero and solve for x.

Relation Between Zeros and Coefficients

• Quadratic Polynomial: (P(x) = ax^2 + bx + c)
• Relation:
  • Sum of Roots: (\alpha + \beta = -\frac{b}{a})
  • Product of Roots: (\alpha \beta = \frac{c}{a})
• Formula: Quadratic Polynomial in terms of roots:
  • (x^2 - (\alpha + \beta)x + \alpha \beta)

Example Problem

• Polynomial: (\sqrt{3}x^2 - 8x + 4\sqrt{3})
  • Set to zero and factorize: ((\sqrt{3}x)(x - 2\sqrt{3}) - 2(x - 2\sqrt{3}) = 0)
  • Zeros: (x = \frac{2}{\sqrt{3}}) and (x = 2\sqrt{3})
  • Verification: Use relations to confirm.
    • Sum: (\alpha + \beta = 8/\sqrt{3})
    • Product: (4)

Finding Coefficients

• If given roots, substitute and solve.
• Example: Roots (x = \frac{2}{3}) and (x = -3)
  • Polynomial: (ax^2 + 7x + b = 0)
  • Substitute roots to form equations.
  • Solve equations to get coefficients a and b.

Important Points

• If a is a zero: ((x - a)) divides (P(x)).
• Example: Zero 2 for (P(x)) implies ((x - 2)) divides (P(x)).

Finding All Zeros

• Cubic Polynomial Example: (2x^3 + x^2 - 6x - 3)
  • Given zeros: (\sqrt{3}) and (-\sqrt{3})
  • Factors: (x - \sqrt{3} ) and (x + \sqrt{3})
  • Find quotient by dividing: (2x + 1)
  • Other zero: Solve (2x + 1 = 0), (x = -\frac{1}{2})

Dividing Polynomials

• Example: Polynomial (x^4 + 2x^3 + 8x^2 + 12x + 18) divided by (x^2 + 5).
  • Quotient: (x^2 + 2x + 3)
  • Remainder: (2x + 3)
  • Coefficients: (P = 2, Q = 3)

Solving for Zeros in Expanded Form

• Example Polynomial: (x^2 + x - p(p+1))
  • Factorize: ((x + p)(x - p) + (x - p) = 0)
  • Zeros: Solve resulting equations.

Final Problem (Challenge)

• Equation: (ax^2 - 6x - 6 = 4)
  • Solve to find values of a and b.

Conclusion

• Apply the techniques learned to solve polynomial questions.
• Practice is key!
• Visit Manoj Academy for more practice.

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Remember to practice these concepts as much as possible, practice makes perfect!