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

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.

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)

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

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

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

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

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

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