Finding All Roots of Polynomials
Working with polynomials is like being a detective - you need strategies to uncover all the roots! Let's put everything together with a systematic approach.
First, use the degree to determine how many roots to look for. A third-degree polynomial will have exactly three roots (counting multiplicity). For example, if we know a cubic passes through points (4,0), (-2,0), and (0,0), we can write it as xx−4x+2 and expand to x³-2x²-8x.
For complete root-finding, follow these steps:
- List all possible rational roots using the Rational Root Theorem
- Test potential roots using a calculator or synthetic division
- When you find a root, use synthetic division to reduce the polynomial
- Solve the remaining "depressed" polynomial for any irrational or complex roots
🔮 Strategy Booster: When a polynomial has an even degree with a positive leading coefficient and a positive constant term, you know it has at least some complex roots sinceitcan′tcrossthex−axisenoughtimesotherwise.
For example, with f(x)=x⁴-4x²-3x+10, list possible rational roots (±1, ±2, ±5, ±10), check them systematically, and when you find one x=−1, use synthetic division to reduce to a cubic. Continue this process until you've found all four roots.