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