Transcript Polynomial Inequalities - Northland Preparatory Academy

POLYNOMIAL INEQUALITIES
They can be written in the forms:
f  x  0 f  x  0
f  x  0
f  x  0
• 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.
f  x    x  3  x  1  x  4 
2
2
Real zeros: x = –3, 4
We can use a sign chart to find the other values of x:
      
Negative
2
–3
      
Positive
      
2
4
Positive
 3, 4  4, 
The function is negative on  , 3
The function is positive on
2
POLYNOMIAL INEQUALITIES
Determine the real number values of x that cause the given
function to be (a) zero, (b) positive, (c) negative.
f  x    x  3  x  1  x  4 
2
2
2
      
      
      
2
2
Negative
–3
Positive
4
Positive
 3, 4  4, 
The function is negative on  , 3
The function is positive 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:
f  x    x  3  x  1  x  4 
2
2
 x  3  x
2
 1  x  4   0
  3,4
 x  3  x
2
 1  x  4   0
  3, 
 x  3  x
2
 1  x  4   0
  , 3
 x  3  x
2
 1  x  4   0
  , 3
2
2
2
2
 4, 
4
POLYNOMIAL INEQUALITIES
Solve the given inequality analytically.
2 x  7 x  10 x  24  0
3
Let
2
f  x   2x3  7x2 10x  24
Use the RZT to list the possible rational zeros!!!
1 3
1, 2, 3, 4, 6, 8, 12, 24,  , 
2 2
Look at the graph  x = 4 seems like a good guess…
POLYNOMIAL INEQUALITIES
Solve the given inequality analytically.
2 x  7 x  10 x  24  0
3
4
2
–7
2
8
1
–10
2
24
4 –24
–6
0
f  x    x  4  2x  x  6
2
f  x    x  4 2x  3 x  2
POLYNOMIAL INEQUALITIES
Solve the given inequality analytically.
2 x  7 x  10 x  24  0
f  x    x  4 2x  3 x  2
3
2
Zeros: 4, 3/2, –2  all of multiplicity 1!!!
Give the end behavior of this function!!!
lim f  x   
x 
lim f  x   
x 
POLYNOMIAL INEQUALITIES
Solve the given inequality analytically.
2 x  7 x  10 x  24  0
f  x    x  4 2x  3 x  2
3
2
Solution:
Now, for the sign chart:
Sign
Change
Neg. due –2
to end
behavior
Positive
Sign
Change
3/2
 2,3 2  4, 
Sign
Change
Negative
4 Pos. due
to end
behavior
x
POLYNOMIAL INEQUALITIES
Solve the given inequality graphically.
x  6 x  2  8x
3
2
Set equal to zero: x  6 x  8 x  2  0
3
2
Rewrite as a function: f  x   x  6x  8x  2
3
2
Graph to find the real zeros and solve the inequality!!!
Solution
  ,0.325
1.461,4.214
POLYNOMIAL INEQUALITIES WITH UNUSUAL
ANSWERS…
Give the solution to each of the following:
 x  7  2 x  1  0  , 
 x  7  2 x  1  0  , 
 x  7  2 x  1  0 No Solution
 x  7  2 x  1  0 No Solution
2
2
2
2
2
2
2
2
The graph???
POLYNOMIAL INEQUALITIES WITH UNUSUAL
ANSWERS…
Give the solution to each of the following:
x
x
x
x
2
The graph???
 3x  3  2 x  5   0  ,  5 2   5 2, 
2
2
 3 x  3  2 x  5   0
 , 
2
 3 x  3  2 x  5   0
No Solution
2
 3 x  3  2 x  5   0
x  5 2
2
2
2