1. Essential Questions 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?

2. Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation.

3. Example 1: Identifying Possible Rational Roots
Use the Rational Root Theorem to identify all possible rational roots.
2x3 – 5x2 – 7x + 21 = 0

4. Example 2: Identifying Possible Rational Roots
Use the Rational Root Theorem to identify all possible rational roots.
5x4 – 7x3 – 15x + 35 = 0

5. Example 3: Identifying all Rational Roots
Use the Rational Root Theorem to identify all rational roots of the equation.
x3 + 3x2 + 3x + 1 = 0
The root -1 has multiplicity 3.

6. Example 4: Identifying all Rational Roots
Use the Rational Root Theorem to identify all rational roots of the equation.
x3 + x2 – 17x + 15 = 0

7. Example 5: Identifying all Rational Roots
Use the Rational Root Theorem to identify all rational roots of the equation.
2x3 – 3x2 – 8x – 3 = 0

8. Polynomial equations may also have irrational roots.

9. The Irrational Root Theorem say that irrational roots come in conjugate pairs. For example, if you know that is a root of x3 – x2 – 3x – 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.

10. Example 6: Identifying All Real Roots of a Polynomial
Identify all the real roots of

11. Example 7: Identifying All Real Roots of a Polynomial
Identify all the real roots of

12. Lesson 4.1 Practice B