Solve graphically (x-2)^3(x+4)(3-x)^2(x+1)^4 <0
You know where the roots are.
You know that if the root is of odd order, the graph crosses the x-axis there.
And, if the root is of even order, the graph is just tangent there.
So, knowing what you do about the general shape of polynomials, since this one is of even order,
The roots are -4,-1,2,3
y > 0 for -∞ < x < -4
y < 0 for -4 < x < -1
y < 0 for -1 < x < 2
y > 0 for 2 < x < 3
y > 0 for 3 < x < ∞
Verify this at
http://www.wolframalpha.com/input/?i=(x-2)%5E3(x%2B4)(3-x)%5E2(x%2B1)%5E4+,+-2+%3C+x+%3C+10%2F3
To solve the inequality graphically, we need to find the values of x for which the expression (x-2)^3(x+4)(3-x)^2(x+1)^4 is less than zero.
To do this, we need to analyze the sign changes of the expression and determine the intervals where it is negative.
First, let's find the critical points by setting each factor equal to zero:
x - 2 = 0 ⟹ x = 2
x + 4 = 0 ⟹ x = -4
3 - x = 0 ⟹ x = 3
x + 1 = 0 ⟹ x = -1
Now, let's create a number line and mark the critical points on it:
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-4 -3 -2 -1 0 1 2 3
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Next, we need to test the sign of the expression in each interval created by the critical points. We can choose a test point in each interval to determine the sign.
For the interval (-∞, -4), we can use x = -5 as the test point. Plugging it into the expression:
(-5-2)^3 (-5+4)(3+5)^2(-5+1)^4 = (-7)^3 * (-1)^1 * (8)^2 * (-4)^4 = -1372 < 0
Since the expression is negative for this interval, we can shade it on the number line.
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-4 -3 -2 -1 0 1 2 3
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(-)
For the interval (-4, -1), we can use x = -2 as the test point. Plugging it into the expression:
(-2-2)^3 (-2+4)(3+2)^2(-2+1)^4 = (0)^3 * (2)^1 * (5)^2 * (-1)^4 = 0 > 0
Since the expression is positive for this interval, we leave it unshaded on the number line.
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-4 -3 -2 -1 0 1 2 3
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(-)
For the interval (-1, 2), we can use x = 0 as the test point. Plugging it into the expression:
(0-2)^3 (0+4)(3-0)^2(0+1)^4 = (-2)^3 * (4)^1 * (3)^2 * (1)^4 = -72 < 0
Since the expression is negative for this interval, we shade it on the number line.
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-4 -3 -2 -1 0 1 2 3
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(-) (-)
For the interval (2, 3), we can use x = 2.5 as the test point. Plugging it into the expression:
(2.5-2)^3 (2.5+4)(3-2.5)^2(2.5+1)^4 = (0.5)^3 * (6.5)^1 * (0.5)^2 * (3.5)^4 = 0.5 * 6.5 * 0.5 * 150.0625 = 48.828125 > 0
Since the expression is positive for this interval, we leave it unshaded on the number line.
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-4 -3 -2 -1 0 1 2 3
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(-) (-)
For the interval (3, ∞), we can use x = 4 as the test point. Plugging it into the expression:
(4-2)^3 (4+4)(3-4)^2(4+1)^4 = (2)^3 * (8)^1 * (-1)^2 * (5)^4 = 8 * 8 * 1 * 625 = 40000 > 0
Since the expression is positive for this interval, we leave it unshaded on the number line.
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-4 -3 -2 -1 0 1 2 3
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(-) (-)
Now, we can see the shaded intervals on the number line where the expression is less than zero:
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-4 -3 -2 -1 0 1 2 3
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(-) (-)
Therefore, the solution to the inequality (x-2)^3(x+4)(3-x)^2(x+1)^4 < 0 is:
-4 < x < -1 and 2 < x < 3.
To solve the inequality graphically, we need to identify the critical points and intervals where the given expression is less than zero. Here's how you can approach it:
1. Identify the critical points:
To find the critical points, set each factor of the expression equal to zero and solve for x.
(x - 2)^3 = 0
x - 2 = 0
x = 2
(x + 4) = 0
x = -4
(3 - x)^2 = 0
3 - x = 0
x = 3
(x + 1)^4 = 0
x + 1 = 0
x = -1
2. Divide the number line:
We divide the number line into intervals based on the critical points we found: -∞, -4, -1, 2, 3, ∞
3. Test the intervals:
Choose a test point in each interval and evaluate the given expression. Check if it is less than zero or not.
Test x = -5 (before -4):
Plug x = -5 into the expression:
(-7)^3(-1)(8)^2(-4)^4 = -343 * -1 * 64 * 256 = -590,643,256
Since this value is negative, the expression is less than zero in this interval.
Test x = -2 (between -4 and -1):
Plug x = -2 into the expression:
(0)^3(2)(5)^2(1)^4 = 0
Since this value is zero, it does not fulfill the condition of less than zero.
Test x = 0 (between -1 and 2):
Plug x = 0 into the expression:
(2)^3(4)(3)^2(2)^4 = 8 * 4 * 9 * 16 = 18,432
Since this value is positive, it does not fulfill the condition of less than zero.
Test x = 2 (between 2 and 3):
Plug x = 2 into the expression:
(0)^3(6)(1)^2(3)^4 = 0
Since this value is zero, it does not fulfill the condition of less than zero.
Test x = 4 (after 3):
Plug x = 4 into the expression:
(2)^3(8)(-1)^2(5)^4 = 8 * 8 * 1 * 625 = 40,000
Since this value is positive, it does not fulfill the condition of less than zero.
4. Determine the intervals where the expression is less than zero:
Based on the test results, we found that the expression is less than zero in the interval: (-∞, -4).
Therefore, the solution to the inequality, (x-2)^3(x+4)(3-x)^2(x+1)^4 < 0, is x < -4.