7.4 Solving Polynomial 7.4 Solving Polynomial EquationsEquations

Objectives: Solve polynomial equations. Objectives: Solve polynomial equations. Find the real zeros of polynomial Find the real zeros of polynomial

functions and state the multiplicity of functions and state the multiplicity of each.each.

Standard: 2.8.11.N. Solve linear, quadratic Standard: 2.8.11.N. Solve linear, quadratic and exponential equations.and exponential equations.

b.b. 22xx 33+ + xx22 – 6– 6xx = 0 = 0

x (2xx (2x22 + x – 6) + x – 6)

x (2x – 3) (x + 2)x (2x – 3) (x + 2)

x = 0, 3/2, and -2x = 0, 3/2, and -2

c.c. 55xx33 – 12– 12xx22 + 4 + 4xx = 0 = 0

x (5xx (5x22 – 12x + 4) – 12x + 4)

x(5x – 2) (x – 2)x(5x – 2) (x – 2)

x = 0, 2/5, and 2x = 0, 2/5, and 2

Some polynomial equations Some polynomial equations have factors (and roots) that have factors (and roots) that

occur more than once.occur more than once.

b.b. xx33+ 2+ 2xx22 – 4 – 4xx – 8 = 0 – 8 = 0

Use a graph, synthetic division, and factoring to find all of the roots of the following polynomial:

c. c. xx33 + 3 + 3xx22 – 4 = 0 – 4 = 0

Use a graph, synthetic division, and factoring to find all of the roots of the following polynomial:

If x – r is a factor that occurs m times in the factorization of a polynomial

expression, P, then r is a root with multiplicity m of

the related polynomial equation, P = 0. In example 2 above, 3 is a root with

multiplicity 2 of the equation x4 - 7x2 + 15x – 9 = 0.

When r is a root with even multiplicity, then the graph of the related function

will touch but not cross the x-axis at (r, 0).

Sometimes polynomials can be factored by using variable substitution.

xx44 – 9x – 9x22 + 14 = 0 + 14 = 0

Use variable substitution and factoring to find all of the roots of the following polynomial:

xx44 – 5x – 5x22 – 6 = 0 – 6 = 0

Use variable substitution and factoring to find all of the roots of the following polynomial:

Writing ActivitiesWriting Activities

Review ofReview ofSolving Polynomial EquationsSolving Polynomial Equations

HomeworkPg. 453 #12-50 even