Polynomial Inequalities

Polynomial Polynomial InequalitiesInequalitiesThe last section in Chapter 2!!!The last section in Chapter 2!!!

Boo Hoo Boo Hoo 2.9a 2.9a

Polynomial InequalitiesPolynomial InequalitiesThey can be written in the forms:

0f x 0f x 0f x 0f x • 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 lookingat the sign of each of its factors…

Polynomial InequalitiesPolynomial InequalitiesDetermine the real number values of x that cause the givenfunction to be (a) zero, (b) positive, (c) negative.

223 1 4f x x x x Real zeros: x = –3, 4

We can use a sign chart to find the other values of x:

–3 4

2 2 2

Negative Positive Positive

The function is positive on 3,4 4, The function is negative on , 3

Polynomial InequalitiesPolynomial InequalitiesDetermine the real number values of x that cause the givenfunction to be (a) zero, (b) positive, (c) negative.

223 1 4f x x x x

–3 4

2 2 2

Negative Positive Positive

The function is positive on 3,4 4, The function is negative on , 3

Does this make sense given the multiplicity of the zeros?

Can we verify all of this graphically ?

Polynomial InequalitiesPolynomial InequalitiesFor the same function, give the solution of each of the following:

223 1 4f x x x x

3,4 4, 223 1 4 0x x x

3, 223 1 4 0x x x

, 3 223 1 4 0x x x

, 3 4 223 1 4 0x x x

Polynomial InequalitiesPolynomial InequalitiesSolve the given inequality analytically.

3 22 7 10 24 0x x x Let 3 22 7 10 24f x x x x

Use the RZT to list the possible rational zeros!!!

1 31, 2, 3, 4, 6, 8, 12, 24, ,

2 2

Look at the graph x = 4 seems like a good guess…

Polynomial InequalitiesPolynomial InequalitiesSolve the given inequality analytically.

3 22 7 10 24 0x x x

4 2 –7 –10 24

8 4 –24

2 1 –6 0

24 2 6f x x x x

4 2 3 2f x x x x

Polynomial InequalitiesPolynomial InequalitiesSolve the given inequality analytically.

3 22 7 10 24 0x x x 4 2 3 2f x x x x

Give the end behavior of this function!!!

limx

f x

limx

f x

Zeros: 4, 3/2, –2 all of multiplicity 1!!!

Polynomial InequalitiesPolynomial InequalitiesSolve the given inequality analytically.

3 22 7 10 24 0x x x 4 2 3 2f x x x x

Now, for the sign chart:

–2

SignChange

Positive Negative x3/2

SignChange

4

SignChange

Pos. dueto end

behavior

Neg. dueto end

behavior

Solution: 2,3 2 4,

Polynomial InequalitiesPolynomial InequalitiesSolve the given inequality graphically.

3 26 2 8x x x 3 26 8 2 0x x x Set equal to zero:

3 26 8 2f x x x x Rewrite as a function:

Graph to find the real zeros and solve the inequality!!!

,0.325 1.461,4.214 Solution

Polynomial Inequalities with Polynomial Inequalities with Unusual Answers…Unusual Answers…

Give the solution to each of the following:

2 27 2 1 0x x ,

2 27 2 1 0x x ,

The graph???

2 27 2 1 0x x

2 27 2 1 0x x

No Solution

No Solution

Polynomial Inequalities with Polynomial Inequalities with Unusual Answers…Unusual Answers…

Give the solution to each of the following:

22 3 3 2 5 0x x x , 5 2 5 2,

The graph???

22 3 3 2 5 0x x x ,

22 3 3 2 5 0x x x No Solution

22 3 3 2 5 0x x x 5 2x