Real Zeros of a Polynomial Function

Objectives: Solve Polynomial Equations.

Apply Descartes RuleFind a polynomial Equation given

the zeros.

Solving Polynomial Functions

• Fundamental Theorem of Algebra:All polynomial functions of degree ‘n’ will have ‘n’ roots.

• Descartes RuleDetermines the possible number of positive and

negative roots by looking at sign changes in the function

Count sign changes in the original function: tells number of maximum positive real roots.

Substitute a negative x in for each x, simplify, then count sign changes: tells number of maximum number of negative real roots.

Determine the number of roots and the possible number of positive and negative roots.

• F(x) = 2x3 + x2 + 2x + 13 roots

No sign changes: no positive real roots

Substitute a negative x in for x and count sign changes

2(-x)3 + (-x)2 + 2(-x) + 1

-2x3 + x2 – 2x + 1: 3 sign changes

3 or 1 negative real root

F(x) = 6x4 – x2 + 2

• 4 roots

• 2 or 0 positive real roots: 2 sign changes

• 6(-x)4 – (-x)2 + 2

• 6x4 – x2 + 2: 2 or 0 negative real roots: 2 sign changes in translated function

Rational Root Theorem

• All rational roots will come from Factors of the last term / factors of the first term

List the potential rational zeros of the polynomial function.

F(x) = 3x5 – x2 + 2x + 18

Factors of the last term: +- 1,2,3,6,9,18

Factors of first term: 1,3

Possible rational roots: 1, 2, 3, 6, 9, 18, 1/3, 2/3, -1, -2, -3, -6, -9, -18, -1/3, -2/3

Solving Polynomial Equations

• 1. Set equal to zero• 2. Use POLY or Graphing to find rational

zeros.• 3. Use synthetic division to get equation

to quadratic form.• 4. Use quadratic formula to solve for

irrational and complex roots• (complex and irrational roots will always

occur in conjugate pairs)

Find the zeros for3x5 + 2x4 + 15x3 + 10x2 – 528x - 352

• 5 answers: at least one positive root

• 4, 2, or 0 negative roots

• Use poly to find “nice” answers

• 4i, -4i, -2/3

• Use synthetic division to get to a quadratic expression then solve for the two remaining solutions using quadratic formula.

F(x) = 3x4 + 5x3 + 25x2 + 45x - 18

• Use POLY

• -3i, 3i, -2, 1/3

• Do not need quadratic formula since all answers came out nice.

Find a function with zeros of 2, 1+i, 2i

• Write all zeros as solutions of x.x = 2, x = 1 + i, x = 1 – i, x = 2i, x = -2i

• Write each in factor formx – 2, x – 1 – i, x – 1 + i, x – 2i, x + 2i

• Multiply factors together (multiply conjugates first)

(x-2)(x-1-i)(x-1+i)(x-2i)(x+2i)

• Product is function with given zeros

Assignment

• Page 374

11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 115

• Page 382

7, 11, 13, 17, 21, 25, 29, 33, 37, 39

Quizzes and test sometime before next Monday.