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Chapter 2: Problem 100

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$

Short Answer

Expert verified
The real zeros of the function are \(z=1/3\), \(z=3/2\), and \(z=-3/2\).

Step by step solution

01

Write down the function

The given function is \(f(z)=12z^{3}-4z^{2}-27z+9\). We need to find all real zeros, i.e., values of z for which \(f(z)=0\).
02

Factor the function

The function can be factored by grouping like terms. This can be achieved by dividing the function into two groups: \(12z^{3}-4z^{2}\) and \(-27z+9\). This results in the factored form: \(4z^{2}(3z-1)-9(3z-1)\). This can be further factored to \((3z-1)(4z^{2}-9)\). The final factorization results in \((3z-1)(2z-3)(2z+3)\).
03

Solve for z

Now, we have three factors: \(3z-1\), \(2z-3\), and \(2z+3\). Each factor is set to zero and solved for z: \n\nFrom \(3z-1=0\), solving for z gives \(z=1/3\).\n\nFrom \(2z-3=0\), solving for z gives \(z=3/2\).\n\nFrom \(2z+3=0\), solving for z gives \(z=-3/2\).

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