Applying the Rational Root Theorem to Find Polynomial Zeros | Precalculus

Understanding the Rational Root Theorem

• 💡 The Rational Root Theorem is a useful tool for finding rational roots (or zeros) of a polynomial function.
• 🎯 It states that if a polynomial has rational roots, they can be expressed as factors of the constant term (P) divided by factors of the leading coefficient (Q).

Identifying Potential Rational Roots

• 🔍 For the function g(x) = 3x³ + 11x² + 5x - 3, the constant term P is -3 and the leading coefficient Q is 3.
• ➕ The factors of P are ±1, ±3. The factors of Q are ±1, ±3.
• ➗ Therefore, the possible rational roots (P/Q) are ±1/1, ±3/1, ±1/3, ±3/3, which simplifies to ±1, ±3, ±1/3.

Testing Potential Roots and Finding Factors

• ✅ Evaluating g(1) resulted in 16, indicating 1 is not a root.
• ✅ Evaluating g(-1) resulted in 0, confirming that x = -1 is a root.
• ➗ Knowing x = -1 is a root implies (x + 1) is a factor. Algebraic long division of (x + 1) into g(x) yields the quadratic factor 3x² + 8x - 3.

Solving the Quadratic Factor

• 🚀 The remaining roots can be found by solving the quadratic equation 3x² + 8x - 3 = 0.
• 📈 Using the quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, with a=3, b=8, and c=-3.
• ⚙️ The calculation yields x = [-8 ± √(64 - 43(-3))] / 6 = [-8 ± √(64 + 36)] / 6 = [-8 ± √100] / 6 = [-8 ± 10] / 6.
• 🌟 This results in two more roots: x = (-8 + 10) / 6 = 2/6 = 1/3, and x = (-8 - 10) / 6 = -18/6 = -3.

Final Roots and Factored Form

• 🔑 The identified roots of the polynomial g(x) are -1, 1/3, and -3.
• 📝 The fully factored form of g(x) is (x + 1)(x - 1/3)(x + 3).