write an expression in factored form for the polynomial of least possible degree graphed below.\n\ny(x) =

Explanation:

Step1: Identify the x-intercepts and their multiplicities

The graph crosses the x-axis at $x = -1$ and $x = 2$ (multiplicity 1) and touches the x-axis at $x = 4$ (multiplicity 2).

Step2: Write the general factored form

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

Step3: Use the y-intercept to find the constant $a$

The graph passes through $(0, -4)$. Substitute $x = 0$ and $y = -4$ into the equation. $$-4 = a(0 + 1)(0 - 2)(0 - 4)^2$$

Step4: Solve for $a$

$$-4 = a(1)(-2)(16) \implies -4 = -32a \implies a = \frac{-4}{-32} = \frac{1}{8}$$

Step5: Write the final expression

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

Answer:

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