Polynomial Inequalities The last section in Chapter 2

Polynomial Inequalities The last section in Chapter 2!!! Boo Hoo 2. 9 a

Polynomial Inequalities They can be written in the forms: • To solve the inequality f (x) > 0 is to find the values of x that make f (x) positive • To solve the inequality f (x) < 0 is to find the values of x that make f (x) negative If the expression f (x) is a product, we can find its sign by looking at the sign of each of its factors…

Polynomial Inequalities Determine the real number values of x that cause the given function to be (a) zero, (b) positive, (c) negative. Real zeros: x = – 3, 4 We can use a sign chart to find the other values of x: Negative – 3 Positive The function is positive on The function is negative on 4 Positive

Polynomial Inequalities Determine the real number values of x that cause the given function to be (a) zero, (b) positive, (c) negative. Negative – 3 Positive 4 Positive The function is positive on The function is negative on Does this make sense given the multiplicity of the zeros? Can we verify all of this graphically ?

Polynomial Inequalities For the same function, give the solution of each of the following:

Polynomial Inequalities Solve the given inequality analytically. Let Use the RZT to list the possible rational zeros!!! Look at the graph x = 4 seems like a good guess…

Polynomial Inequalities Solve the given inequality analytically. 4 2 2 – 7 8 1 – 10 24 4 – 24 – 6 0

Polynomial Inequalities Solve the given inequality analytically. Zeros: 4, 3/2, – 2 all of multiplicity 1!!! Give the end behavior of this function!!!

Polynomial Inequalities Solve the given inequality analytically. Solution: Now, for the sign chart: Sign Change Neg. due – 2 to end behavior Positive Sign Change 3/2 Sign Change Negative 4 Pos. due to end behavior x

Polynomial Inequalities Solve the given inequality graphically. Set equal to zero: Rewrite as a function: Graph to find the real zeros and solve the inequality!!! Solution

Polynomial Inequalities with Unusual Answers… Give the solution to each of the following: No Solution The graph? ? ?

Polynomial Inequalities with Unusual Answers… Give the solution to each of the following: The graph? ? ? No Solution