Fill in the rest of the table to describe the polynomial at it’s x-intercepts. x2(3x+6)3(2x-4)(x+15)4

Answer 1

To analyze the polynomial and describe its behavior at the x-intercepts, we first need to identify the x-intercepts and their multiplicities from the given polynomial expression: \[ P(x) = x^2(3x + 6)^3(2x - 4)(x + 15)^4 \]

Now let's break this polynomial down:

1. Find the x-intercepts:

   • \( x^2 = 0 \) gives \( x = 0 \) with multiplicity 2.
   • \( (3x + 6)^3 = 0 \) gives \( 3x + 6 = 0 \) or \( x = -2 \) with multiplicity 3.
   • \( (2x - 4) = 0 \) gives \( 2x - 4 = 0 \) or \( x = 2 \) with multiplicity 1.
   • \( (x + 15)^4 = 0 \) gives \( x + 15 = 0 \) or \( x = -15 \) with multiplicity 4.

2. List the x-intercepts and their multiplicities:

   • \( x = 0 \) with multiplicity 2
   • \( x = -2 \) with multiplicity 3
   • \( x = 2 \) with multiplicity 1
   • \( x = -15 \) with multiplicity 4

3. Behavior at the x-intercepts:

   • For a x-intercept with odd multiplicity, the graph crosses the x-axis at that intercept.
   • For a x-intercept with even multiplicity, the graph touches the x-axis and turns around.

Now we can summarize this information in a table:

| x-intercept | Multiplicity | Behavior at the x-intercept | |--------------|--------------|--------------------------------------| | 0 | 2 | Touches the x-axis and turns around | | -2 | 3 | Crosses the x-axis | | 2 | 1 | Crosses the x-axis | | -15 | 4 | Touches the x-axis and turns around |

This table effectively summarizes the x-intercepts, their multiplicities, and the behavior of the polynomial at each of the x-intercepts.