Questions: For the polynomial below, -2 is a zero. g(x)=x^3+3 x^2-4 x-12 Express g(x) as a product of linear factors. g(x)=

For the polynomial below, -2 is a zero. g(x)=x^3+3 x^2-4 x-12 Express g(x) as a product of linear factors. g(x)=
Transcript text: For the polynomial below, -2 is a zero. \[ g(x)=x^{3}+3 x^{2}-4 x-12 \] Express $g(x)$ as a product of linear factors. \[ g(x)= \]
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Solution

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Solution Steps

Step 1: Identify a known zero

The given known zero is \(r = -2\).

Step 2: Perform synthetic division

After synthetic division, the coefficients of the quadratic polynomial are \(a_2 = 1\), \(b_2 = 1\), and \(c_2 = -6\).

Step 3: Find the roots of the resulting quadratic polynomial

The roots of the quadratic polynomial are \(s = 2\) and \(t = -3\).

Step 4: Express the original polynomial as a product of linear factors

The original cubic polynomial can be expressed as \(p(x) = 1(x + 2)(x - 2)(x + 3)\).

Final Answer:

\(p(x) = 1(x + 2)(x - 2)(x + 3)\)

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