Algebra

Complex Roots in Polynomials

• Imaginary roots of polynomial equations with real coefficients occur in conjugate pairs.

  • If ( a + b i ) is a root, then ( a - b i ) is also a root.

  • Example: Given one complex root, the other is automatically identified.

Finding Possible Rational Zeros

• Utilize the Irrational Zero Theorem to identify potential rational zeros.

  • Constant Term: 13

    • Factors of 13: ( \pm 1, \pm 13 )

  • Leading Coefficient: 1

    • Factors of 1: ( \pm 1 )

  • Possible Rational Zeros: ( \pm 1, \pm 13 )

Checking for Zeros

• To determine if a value is a zero, plug it into the polynomial:

  • Example with 1:

    • Calculation: 17 (previous value) + (-30) = -13 (not zero)

  • Value: 1 is confirmed as a zero because it led to zero.

Factoring the Polynomial

• Once a zero is discovered, it can be expressed as a linear factor: