Finding Zeros of a Polynomial Function

Introduction

• The goal is to find the zeros of a polynomial function, which are values of x that make the polynomial equal to zero.

Rational Zero Theorem

• The Rational Zero Theorem helps to identify possible rational zeros of a polynomial function.
• It states that the possible zeros can be expressed as ( \frac{p}{q} ), where:
  • ( p ): factors of the constant term
  • ( q ): factors of the leading coefficient

Example Polynomial Function

• Given polynomial function: ( p(x) = x^3 + 2x^2 - 5x - 6 ) (cubic polynomial)

Step 1: Identify Factors

• Factors of the constant term (-6):
  • 1, -1, 2, -2, 3, -3, 6, -6
  • Simplified: ( { \pm 1, \pm 2, \pm 3, \pm 6 } )
• Factors of the leading coefficient (1):
  • 1, -1
  • Simplified: ( { \pm 1 } )

Step 2: List Possible Zeros

• Possible zeros calculated as ( \frac{p}{q} ):
  • Possible zeros: ( { \pm 1, \pm 2, \pm 3, \pm 6 } )

Step 3: Testing Possible Zeros

1. Test x = 1:

   • Compute ( p(1) = 1^3 + 2(1^2) - 5(1) - 6 )
   • Result: ( p(1) = -8 ) (not a zero)

2. Test x = -1:

   • Compute ( p(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 )
   • Result: ( p(-1) = 12 ) (not a zero)

3. Test x = 2:

   • Compute ( p(2) = 2^3 + 2(2^2) - 5(2) - 6 )
   • Result: ( p(2) = 0 ) (is a zero)

Step 4: Synthetic Division

• Use synthetic division to further factor the polynomial:
  • Coefficients: ( [1, 2, -5, -6] )
  • After synthetic division with zero found (x=2), we get a quadratic polynomial:
  • New polynomial: ( x^2 + 4x + 3 )

Step 5: Factoring Quadratic Polynomial

• Factor ( x^2 + 4x + 3 = 0 ):
  • Factors: ( (x + 3)(x + 1) = 0 )
  • Zeros found: ( x = -3, x = -1 )

Summary of Zeros

• The zeros of the polynomial function are:
  • ( x = 2, x = -3, x = -1 )

Example 2: Degree 4 Polynomial Function

• Given function: ( f(x) = x^4 - 5x^3 + 20x - 16 )

Step 1: Identify Factors

• Factors of constant term (-16):
  • ( { \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 } )
• Leading coefficient (1):
  • ( { \pm 1 } )

Step 2: List Possible Zeros

• Possible zeros: ( { \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 } )

Step 3: Testing Possible Zeros

• Test positive 1 using synthetic division:
  • Result shows remainder is zero, hence ( x = 1 ) is a zero.

Step 4: Factoring Remaining Polynomial

1. Resulting polynomial after synthetic division:
   • ( x^3 - 4x^2 - 4x + 16 )
2. Factor by grouping:
   • Resulting factors: ( (x - 1)(x + 2)(x - 2)(x - 4) )

Summary of Zeros

• The zeros of the degree 4 polynomial function are:
  • ( x = 1, x = -2, x = 2, x = 4 )

Conclusion

• Understanding how to find zeros of polynomial functions is crucial.
• The use of the Rational Zero Theorem, synthetic division, and factoring are key methods to achieve this.