(x-3)%5E2(x-7)%5E3-0.jpeg "(x+8)(x-3)^2(x-7)^3 0")
Solving the Polynomial Inequality: (x+8)(x-3)^2(x-7)^3 < 0
This article will guide you through solving the polynomial inequality (x+8)(x-3)^2(x-7)^3 < 0. We'll break down the steps involved, using the concepts of zeros, multiplicity, and sign analysis.
1. Find the Zeros
The zeros of the polynomial are the values of x that make the expression equal to zero. To find these, set each factor equal to zero and solve:
- x + 8 = 0 => x = -8
- (x - 3)^2 = 0 => x = 3
- (x - 7)^3 = 0 => x = 7
2. Determine the Multiplicity of Each Zero
The multiplicity of a zero is the number of times it appears as a root in the factored polynomial.
- x = -8 has a multiplicity of 1 (appears once)
- x = 3 has a multiplicity of 2 (appears twice)
- x = 7 has a multiplicity of 3 (appears three times)
3. Create a Sign Chart
A sign chart helps us visualize the intervals where the polynomial is positive or negative.
-
Number Line: Draw a number line and mark the zeros (-8, 3, and 7) on it. These points divide the number line into four intervals:
- x < -8
- -8 < x < 3
- 3 < x < 7
- x > 7
-
Test Points: Choose a test value within each interval and substitute it into the original polynomial. The sign of the result tells us the sign of the polynomial in that interval.
| Interval | Test Value | Polynomial Value | Sign |
|---|---|---|---|
| x < -8 | x = -9 | (-1)(-12)^2(-16)^3 | - |
| -8 < x < 3 | x = 0 | (8)(-3)^2(-7)^3 | + |
| 3 < x < 7 | x = 5 | (13)(2)^2(-2)^3 | - |
| x > 7 | x = 8 | (16)(5)^2(1)^3 | + |
- Fill in the Sign Chart: Based on the test values, fill in the sign chart with "+" or "-" to indicate whether the polynomial is positive or negative in each interval.
4. Solve the Inequality
The inequality (x+8)(x-3)^2(x-7)^3 < 0 means we're looking for the intervals where the polynomial is negative.
Solution:
The polynomial is negative in the following intervals:
- x < -8
- 3 < x < 7
In interval notation, the solution is:
- (-∞, -8) U (3, 7)
Important Note:
- Since the inequality is strict (<), the zeros with odd multiplicities (x = -8 and x = 7) are not included in the solution.
- The zero with even multiplicity (x = 3) is included in the solution because it doesn't change the sign of the polynomial.
Remember to always check your solutions by plugging in test values from each interval into the original inequality.