Aim: What are the higher degree function and equation?

Do Now: a) Graph f(x) = x3 + x2 – x – 1 on the calculator

b) How many times does the graph intersect the x-axis?

HW: Worksheet

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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The intersecting points of the graph and the x-axis are called the zeros of the function, because at those points the value of y is 0.

f(x) = x3 + x2 – x – 1

The zeros of the function f(x) = x3 + x2 – x –1 is just like the root of the equation x3 + x2 – x – 1 = 0

Why are there only two zeros for a third degree function?

We can tell the zeros are 1 and –1 from the graph

The graph is tangent to the x-axis at x = – 1, therefore x = –1 is a repeated zeros (double roots)

To find the zeros (roots) of a function, we can use either graphical or algebraic methods.

Graphically:

find the x-values of the intersecting points of the graph of the function and the x-axis.

Algebraically:

Let f(x) = 0, then solve the variable

Ex: x3 + x2 – x – 1 = 0

Regrouping: (x3 + x2) – (x + 1) = 0

Factor: x2(x + 1) – (x + 1) = 0

Factor GCF: (x + 1)(x2 – 1) = 0

x + 1 = 0

x = –1

012 x

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a. Graph: f(x) = x4 – 3x2 + 2b. Solve: x4 – 3x2 + 2 = 0Math Composer 1. 1. 5

http: / /www. mathcomposer. com

-5 -4 -3 -2 -1 1 2 3 4 5

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y f(x) = x4 – 3x2 + 2

023 24 xx

We can treat this equation as a quadratic equation

023)( 222 xx

Factor: 0)1)(2( 22 xxSet each binomial equals zero

x2 – 2 = 0 x2 – 1 = 0

Solve for x: ,22 x 2x

,12 x 1x

x2(x2 4x 4) 0 Factor out x2.

x2(x 2)2 0 Factor completely.

x2 or (x 2)2 0 Set each factor equal to zero.

x 0 x 2 Solve for x.

x4 4x3 4x2 0 Multiply both sides by 1. (optional step)

Solve for x: x4 4x3 4x2 = 0

Solve equation by grouping

Solve for x: x4 – x3 + x – 1 = 0

(x4 – x3) + (x – 1) = 0 Grouping as two binomials

x3(x – 1) + (x – 1) = 0 Factor x3

(x – 1)(x3 + 1) = 0 Factor (x – 1)

x – 1 = 0 x3 + 1 = 0 Set each binomial equals zero

x = 1 x = -1 Solve for x

Solve for x:

1. x3 – 2x2 – 3x = 0

2. x4 + 5x2 – 36 = 0

3. 5x3 + 30x2 + 45x = 0

4. 32x3 – 16x2 – 18x + 9 = 0

Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula and / graphically

Objective

Students will be able to

1. define the degree of a polynomial equation2. factor polynomial expressions of degree > 33. identify polynomials that are written in ‘quadratic form’4. state and apply the quadratic formula5. express irrational solutions in simplest radical form6. graphically identify (estimate) x-intercepts as solution of a polynomial equation7. graphically identify (estimate) the x-coordinate of the point of intersection of a system of polynomial equations as a solution of that

system

Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)

Use multiple representations to represent and explain problem situations

Use a variety of strategies to extend solution methods to other problems

Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions

Evaluate the relative efficiency of different representations and solution methods of a problem

Performance Standard