Finding Real Roots of Polynomial Equations How do

Finding Real Roots of Polynomial Equations • How do we identify the multiplicity of roots? • How do we use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations? Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation. Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 1: Identifying Possible Rational Roots Use the Rational Root Theorem to identify all possible rational roots. 2 x 3 – 5 x 2 – 7 x + 21 = 0 Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 2: Identifying Possible Rational Roots Use the Rational Root Theorem to identify all possible rational roots. 5 x 4 – 7 x 3 – 15 x + 35 = 0 Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 3: Identifying all Rational Roots Use the Rational Root Theorem to identify all rational roots of the equation. x 3 + 3 x 2 + 3 x + 1 = 0 The root -1 has multiplicity 3. Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 4: Identifying all Rational Roots Use the Rational Root Theorem to identify all rational roots of the equation. x 3 + x 2 – 17 x + 15 = 0 Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 5: Identifying all Rational Roots Use the Rational Root Theorem to identify all rational roots of the equation. 2 x 3 – 3 x 2 – 8 x – 3 = 0 Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Polynomial equations may also have irrational roots. Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations The Irrational Root Theorem say that irrational roots come in conjugate pairs. For example, if you know that 1 + is a root of x 3 – x 2 – 3 x – 1 = 0, then you know that 1 – is also a root. Recall that the real numbers are made up of the rational and irrational numbers. You can use the Rational Root Theorem and the Irrational Root Theorem together to find all of the real roots of P(x) = 0. Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 6: Identifying All Real Roots of a Polynomial Identify all the real roots of Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Example 7: Identifying All Real Roots of a Polynomial Identify all the real roots of Holt Mc. Dougal Algebra 2

Finding Real Roots of Polynomial Equations Lesson 4. 1 Practice B Holt Mc. Dougal Algebra 2