Q. The function g is defined by \(g(x)=x(x-2)(x+6)^2\).

Problem: Analyze and Solve the function \(g(x) = x(x – 2)(x + 6)^2\)

The goal is to analyze the properties of this polynomial function, including its roots, intercepts, and expansion.

1. Step 1 – Identify the roots (x-intercepts):

   To find the roots, we set the function equal to zero: \(x(x – 2)(x + 6)^2 = 0\).

   Using the Zero Product Property, we solve for each factor separately:

   • \(x = 0\)
   • \(x – 2 = 0 \implies x = 2\)
   • \((x + 6)^2 = 0 \implies x = -6\)

   The roots are \(x = 0\), \(x = 2\), and \(x = -6\).

2. Step 2 – Determine the multiplicity of each root:

   The multiplicity tells us how the graph behaves at the x-axis:

   • \(x = 0\) has a multiplicity of \(1\) (odd), so the graph crosses the x-axis.
   • \(x = 2\) has a multiplicity of \(1\) (odd), so the graph crosses the x-axis.
   • \(x = -6\) has a multiplicity of \(2\) (even), so the graph touches the x-axis and bounces back.
3. Step 3 – Find the y-intercept:

   To find the y-intercept, evaluate the function at \(x = 0\):

   \(g(0) = 0(0 – 2)(0 + 6)^2\)

   \(g(0) = 0\cdot(-2)\cdot 36 = 0\)

   The y-intercept is \((0, 0)\).

4. Step 4 – Expand the polynomial expression:

   First, expand the squared term: \((x + 6)^2 = (x + 6)(x + 6) = x^2 + 12x + 36\).

   Next, multiply the first two factors: \(x(x – 2) = x^2 – 2x\).

   Now, multiply these two results together:

   \((x^2 – 2x)(x^2 + 12x + 36)\)

   \(= x^2(x^2 + 12x + 36) – 2x(x^2 + 12x + 36)\)

   \(= x^4 + 12x^3 + 36x^2 – 2x^3 – 24x^2 – 72x\)

   Combine like terms:

   \(g(x) = x^4 + 10x^3 + 12x^2 – 72x\)

5. Step 5 – Determine end behavior:

   The leading term is \(x^4\). Since the degree (\(4\)) is even and the leading coefficient (\(1\)) is positive, the end behavior is:

   As \(x \to \infty,\, g(x) \to \infty\)

   As \(x \to -\infty,\, g(x) \to \infty\)

6. Final Summary:

   The function \(g(x) = x(x – 2)(x + 6)^2\) is a fourth-degree polynomial with roots at \(x = -6\) (touch), \(x = 0\) (cross), and \(x = 2\) (cross). In standard form, the function is written as \(g(x) = x^4 + 10x^3 + 12x^2 – 72x\).