Question

write an expression in factored form for the polynomial

\\(y(x) =\\)

Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Identify the roots and their multiplicities

Identify the \(x\)-intercepts (roots) from the graph and observe how the curve behaves at each intercept:

• At \(x = -4\), the graph crosses the \(x\)-axis directly. This indicates a root of multiplicity 1: factor \((x + 4)\).
• At \(x = -3\), the graph crosses the \(x\)-axis directly. This indicates a root of multiplicity 1: factor \((x + 3)\).
• At \(x = 1\), the graph touches the \(x\)-axis and turns around. This indicates a root of multiplicity 2: factor \((x - 1)^2\).
• At \(x = 3\), the graph crosses the \(x\)-axis directly. This indicates a root of multiplicity 1: factor \((x - 3)\).

The general factored form is:

$$ y(x) = a(x + 4)(x + 3)(x - 1)^2(x - 3) $$

Step 2: Find the leading coefficient \(a\)

Use the \(y\)-intercept from the graph, which is at \((0, -4.5)\) or \((0, -\frac{9}{2})\):

$$ y(0) = a(0 + 4)(0 + 3)(0 - 1)^2(0 - 3) = -4.5 $$

$$ a(4)(3)(1)(-3) = -4.5 $$

$$ -36a = -4.5 $$

$$ a = \frac{-4.5}{-36} = \frac{1}{8} $$

Step 3: Write the final expression

Substitute \(a = \frac{1}{8}\) back into the factored form:

$$ y(x) = \frac{1}{8}(x + 4)(x + 3)(x - 1)^2(x - 3) $$

Answer:

$$ y(x) = \frac{1}{8}(x + 4)(x + 3)(x - 1)^2(x - 3) $$