Can you help me with this problem?
Use the intermediate value theorem to determine whether
g(x)=4x^3-3x+3 has a zero between -2 and -1
We first find g(-2) by either of the two methods:
1. Direct substitution
g(-2) = 4(-2)³-3(-2)+3 = 4(-8)+6+3 = -23
Or
2. By using the remainder theorem and synthetic
division.
Write g(x) = 4x³ - 3x + 3 as
g(x) = 4x³ + 0x² - 3x + 3
Then use synthetic division with -2
-2 | 4 0 -3 3
| -8 16 -26
4 -8 13 -23
And the remainder, the last number
on the bottom of the synthetic division,
which is -23, is the value of f(-2)
You can use whichever method to find
f(-2) = -23 your teacher said to use.
Next we first find g(-1) by either of those same
two methods:
1. Direct substitution
g(-1) = 4(-1)³-3(-1)+3 = 4(-1)+3+3 = 2
Or, again,
2. By using the remainder theorem and synthetic
division.
Write g(x) = 4x³ - 3x + 3 as
g(x) = 4x³ + 0x² - 3x + 3
Then use synthetic division with -1
-1 | 4 0 -3 3
| -4 4 -1
4 -4 -1 2
And the remainder, the last number
on the bottom of the synthetic division,
which is 2, is the value of f(-1)
Again, you can use whichever method to find
f(-1) = 2 your teacher said to use.
Now you notice that f(-2) = -23 and
f(-1) = +2. Notice that one of them is a
negative number and the other is a positive
number. This means that some number between
x = -2 and x = -1, the function's graph had
to cross the x axis. Therefore we know that
there is at least one zero between -2 and -1.
Take a look at the graph of g(x) = 4x³ - 3x + 3:
You can see that the graph goes through (-2,-23)
and (-1,2) and so it must cross the x-axis
between them. It appears to cross at about -1.2
and that is a zero between -2 and -1.
Edwin
