The Real Roots of Polynomials 4 November 2010

Finding the Roots for Polynomials If given one or more of the roots to a polynomial, we can solve for the remaining roots using synthetic division and factoring.

Finding the Roots for Polynomials, cont. Given a cubic, then you need: 1 root Synthetic division once Factor once Given a quartic, then you need: 2 roots Synthetic division twice

Finding the Roots for Polynomials, cont. For Cubic Equations: Step 1: Convert the root into a linear factor (if necessary) Step 2: Use synthetic division to solve for a second factor. (It will probably be a quadratic equation.) Step 3: Factor the quadratic equation. Step 4: Solve for x.

Finding the Roots for Polynomials, cont. x3 – 7x + 6 = 0 x = 2

Finding the Roots for Polynomials, cont. For Quartic Equations: Step 1: Convert both roots into linear factors (if necessary) Step 2: Use synthetic division with one of the linear factors. Step 3: Use synthetic division on the quotient from step 2 with the other linear factor. Step 4: Factor the result of step 3. Step 5: Solve for x.

Finding the Roots for Polynomials, cont. 2x4 + 7x3 – 4x2 – 27x – 18 x = 2 x = -3

Finding the Roots for Polynomials, cont. 2x4 + 7x3 – 4x2 – 27x – 18 x = 2 x = -3

*Your Turn: Complete problems 1 – 4 on “The Real Roots of Polynomials Practice” Handout. USE ONLY SYNTHETIC DIVISION AND FACTORING!!!!

Using Your Graphing Calculator to Find the Roots Once the equation has been graphed: 2nd Trace (Calc) 2 (for zero) Set Left Bound Set Right Bound Set Guess

Using Your Graphing Calculator to Find the Roots, cont. Ex. y = x4 – 3x3 + x2 – 5

Your Turn: Use your calculator to check the roots that you found on “The Real Roots of Polynomials” handout.

Why can’t we just use our graphing calc. to find the roots? Because sometimes there are roots far outside standard viewing windows. Ex. y = x3 – 53x2 + 10x – 51

Homework “The Real Roots of Polynomials Homework” Handout: 5 – 8