(x+2)^3(X-1)^4(x+3)^2
Write this solution in interval notation.
Can someone explain how to do the method where you start with a + on the number line, and depending on the odd or even value of the next exponent, you switch to and -/+ ?
To write the solution in interval notation, we need to determine the intervals for which the expression is positive or negative. We can use the method you mentioned, which is called the "sign chart" method. Here's how you can do it step by step:
1. Factor the given expression completely: (x + 2)^3 * (x - 1)^4 * (x + 3)^2.
2. Determine the critical points by setting each factor equal to zero and solving for x. In this case, the critical points are -2, 1, and -3.
3. Create a sign chart by writing the critical points in ascending order on a number line. Place a "+" sign above each region on the number line where the corresponding factor is positive, and place a "-" sign where the factor is negative. I'll explain how to determine the signs based on the exponents.
- Start with a "+" sign above the first region (leftmost) on the number line, since x = -∞ is negative.
- For the factor (x + 2)^3, the exponent is odd (3), so the sign alternates starting with a "+" sign, then a "-" sign, then a "+" sign, and so on.
- For the factor (x - 1)^4, the exponent is even (4), so the sign remains positive throughout all regions.
- For the factor (x + 3)^2, the exponent is even (2), so the sign remains positive throughout all regions.
- The last region (rightmost) will have a "+" sign.
4. Write the solution in interval notation based on the sign chart.
- For regions where the sign is positive, we include the interval in the solution.
- For regions where the sign is negative, we exclude the interval from the solution.
In this case, the solution in interval notation is:
(-∞, -3) U (-2, 1) U (1, ∞)
This includes all the intervals where the expression (x + 2)^3 * (x - 1)^4 * (x + 3)^2 is positive, and excludes the intervals where it is negative.