Question

write the equation of the polynomial below.

write the equation of the polynomial by writing all the factors:

\\(p(x) = \\)

Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Identify the zeros and their multiplicities

From the graph, we observe the behavior of the polynomial \( p(x) \) at its \( x \)-intercepts (zeros):

• At \( x = 2 \), the graph touches the \( x \)-axis and turns around. This indicates a zero of even multiplicity, typically \( 2 \).
• Factor: \( (x - 2)^2 \)

• At \( x = 3 \), the graph crosses the \( x \)-axis while flattening out (inflection point). This indicates a zero of odd multiplicity greater than 1, typically \( 3 \).
• Factor: \( (x - 3)^3 \)

Step 2: Identify the vertical asymptotes

The graph has vertical asymptotes, which means this is a rational function rather than a standard polynomial. Let's identify the locations of these vertical asymptotes:

• There is a vertical asymptote at \( x = -3 \).
• Factor in the denominator: \( (x + 3) \)

• There is a vertical asymptote at \( x = -1 \).
• Factor in the denominator: \( (x + 1) \)

Since the graph goes in opposite directions on either side of each asymptote (from \( -\infty \) to \( +\infty \)), these asymptotes have an odd multiplicity, typically \( 1 \).

Step 3: Write the general form of the function

Combining the factors of the numerator (zeros) and the denominator (asymptotes):

$$ p(x) = a \cdot \frac{(x - 2)^2 (x - 3)^3}{(x + 3)(x + 1)} $$

Step 4: Determine the leading coefficient

Let's find a clear point on the graph to determine the constant \( a \).

Looking at the \( y \)-intercept where \( x = 0 \):
The graph passes through \( (0, 12) \). Let's substitute \( x = 0 \) and \( p(0) = 12 \) into our equation:

$$ 12 = a \cdot \frac{(0 - 2)^2 (0 - 3)^3}{(0 + 3)(0 + 1)} $$

$$ 12 = a \cdot \frac{(4)(-27)}{(3)(1)} $$

$$ 12 = a \cdot \frac{-108}{3} $$

$$ 12 = -36a $$

$$ a = -\frac{12}{36} = -\frac{1}{3} $$

Step 5: Write the final equation

Substituting \( a = -\frac{1}{3} \) back into the function:

$$ p(x) = -\frac{(x - 2)^2 (x - 3)^3}{3(x + 3)(x + 1)} $$

Answer:

$$ p(x) = -\frac{(x - 2)^2 (x - 3)^3}{3(x + 3)(x + 1)} $$