Overview
This lecture covers solving polynomial factorization problems using synthetic division, with a focus on finding linear factors and confirming results graphically.
Synthetic Division with Non-Integer Zeros
- When given a zero (e.g., √2), use it directly in synthetic division (no sign change).
- Write polynomial coefficients in order and use the zero outside the synthetic division setup.
- Perform synthetic division by dropping the leading coefficient, multiplying by the zero, and adding down the column.
- The result gives a quadratic factor, which can be further factored into linear factors if possible.
Factoring Quadratics into Linear Factors
- A quadratic factor from synthetic division may be factorable into two linear factors.
- Linear factors have the form (x + a) or (x - a), i.e., x to the first power.
- To factor quadratics: find two numbers that multiply to the constant term and add to the coefficient of the x-term.
Repeated Synthetic Division for Higher-Degree Polynomials
- If given multiple factors, apply synthetic division repeatedly to reduce the degree of the polynomial.
- Each successful division by a root (zero) reduces the polynomial's degree by one.
- Multiple synthetic divisions can turn an x⁴ polynomial into an x² polynomial.
- The process works in any order for the given zeros.
Compiling Complete Factorization & Graphical Verification
- Combine all found linear factors for the complete factorization.
- Example complete factorization: (x + 2)(x - 4)(4x + 3)(2x - 1).
- Check your factorization by expanding and verifying it matches the original polynomial.
- Confirm zeros using a graphing calculator—graph the polynomial and verify roots visually.
Key Terms & Definitions
- Synthetic Division — a shortcut method for dividing polynomials by linear factors.
- Zero — a value x = a for which the polynomial equals zero (root of the equation).
- Linear Factor — a factor in the form (x - a), representing a root.
- Quadratic Factor — a second-degree polynomial factor (e.g., x² + bx + c).
Action Items / Next Steps
- Practice synthetic division with given zeros and factor resulting quadratics.
- Use a graphing calculator to verify polynomial roots visually.
- Complete the synthetic division quiz or section 2.3 quiz as assigned.