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Chapter 3: Problem 54

If you know that \(-2\) is a zero of $$f(x)=x^{3}+7 x^{2}+4 x-12$$ explain how to solve the equation $$x^{3}+7 x^{2}+4 x-12=0$$

Short Answer

Expert verified
By confirmation, -2 is a root of the function. Use -2 to divide the function via synthetic division to simplify the function to a quadratic form. Solving this resulting quadratic will yield the remaining solutions to the original equation.

Step by step solution

01

Confirm the given zero

Plug -2 into the given function: \(f(-2) = (-2)^3 + 7(-2)^2 + 4(-2) - 12\). Simplify this equation and if the result is zero, -2 is indeed a root of the function.
02

Use Synthetic Division to find the quadratic factor

Since we verified that -2 is a zero of the cubic function, we can use it to divide the function \(f(x)=x^{3}+7 x^{2}+4 x-12\) via synthetic division to get a quadratic equal to zero.
03

Find Quadratic Roots

The result from step 2 will provide a quadratic equation in the form \(ax^2 + bx + c = 0\). Solve this quadratic equation for its roots, which are the remaining solutions to the original cubic equation.

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