1. HW: Worksheet Aim: What are the higher degree function and equation?
Do Now: a) Graph y = x3 + x2 – x – 1
b) How many times does the graph intersect the x-axis?
HW: Worksheet

2. y = x3 + x2 – x – 1
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.

3. The zeros of the equation y = x3 + x2 – x –1 is just like the root of the equation x3 + x2 – x – 1 = 0
We can tell the zeros are 1 and –1 from the graph
Why are there only two zeros for a third degree function?
The graph is tangent to the x-axis at x = – 1, therefore x = –1 is a repeated zeros (double roots)

4. To find the zeros (roots) of an equation, 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

5. 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

6. Graph: f(x) = x4 – 3x2 + 2
Solve: x4 – 3x2 + 2 = 0
f(x) = x4 – 3x2 + 2

7. We can treat this equation as a quadratic equation
Factor:
Set each binomial equals zero
x2 – 2 = x2 – 1 = 0
Solve for x:

8. x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero.
Solve for x: -x4 + 4x3 - 4x2 = 0
x4 - 4x3 + 4x2 = Multiply both sides
by -1. (optional step)
x2(x2 - 4x + 4) = Factor out x2.
x2(x - 2)2 = Factor completely.
x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero.
x = x = Solve for x.

9. 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 = x3 + 1 = 0
Set each binomial equals zero
x = 1
x = -1
Solve for x

10. Solve for x: 1. x3 – 2x2 – 3x = 0 2. x4 + 5x2 – 36 = 0