- Identify potential rational zeros using the Rational Root Theorem.
- Test potential roots to find the first zero: \(x = \frac{1}{3}\).
- Factor the polynomial by dividing by \((3x - 1)\) to get the quadratic equation \(3x^2 + 4x - 4 = 0\).
- Solve the quadratic equation to find the final two zeros: \(x = \frac{2}{3}\) and \(x = -2\).
- The complete set of zeros is \(\boxed{{-2, \frac{1}{3}, \frac{2}{3}}}\).
Explanation
- Analysis and Rational Root Theorem
We are given the cubic polynomial \(h(x) = 9x^3 + 9x^2 - 16x + 4\) and informed that it has at least one rational zero. To find all the zeros of this polynomial, we can employ the Rational Root Theorem. According to this theorem, any rational root of a polynomial \(a_n x^n + \dots + a_0\) must be in the form \(\frac{p}{q}\), where \(p\) is a divisor of the constant term \(a_0 = 4\) and \(q\) is a divisor of the leading coefficient \(a_n = 9\). The possible values for \(p\) are \(\{\pm 1, \pm 2, \pm 4\}\) and the possible values for \(q\) are \(\{1, 3, 9\}\). Thus, potential rational zeros are \(\{\pm 1, \pm 2, \pm 4, \pm 1/3, \pm 2/3, \pm 4/3, \pm 1/9, \pm 2/9, \pm 4/9\}\).
- Identifying a Rational Root
We test the potential rational zeros by evaluating \(h(x)\). Testing \(x = 1/3\):
\[h(1/3) = 9\left(\frac{1}{3}\right)^3 + 9\left(\frac{1}{3}\right)^2 - 16\left(\frac{1}{3}\right) + 4\]
\[h(1/3) = 9\left(\frac{1}{27}\right) + 9\left(\frac{1}{9}\right) - \frac{16}{3} + 4\]
\[h(1/3) = \frac{1}{3} + 1 - \frac{16}{3} + 4 = 5 - \frac{15}{3} = 5 - 5 = 0.\]
Since \(h(1/3) = 0\), \(x = 1/3\) is a rational zero of the function. This implies that \((x - 1/3)\) or \((3x - 1)\) is a factor of the polynomial.
- Finding All Zeros
To find the remaining roots, we divide \(h(x) = 9x^3 + 9x^2 - 16x + 4\) by the factor \((3x - 1)\). Performing polynomial division:
\[(9x^3 + 9x^2 - 16x + 4) \div (3x - 1) = 3x^2 + 4x - 4\]
Now, we solve the quadratic equation \(3x^2 + 4x - 4 = 0\) to find the remaining two roots. We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=3, b=4, c=-4\):
\[x = \frac{-4 \pm \sqrt{4^2 - 4(3)(-4)}}{2(3)} = \frac{-4 \pm \sqrt{16 + 48}}{6} = \frac{-4 \pm \sqrt{64}}{6} = \frac{-4 \pm 8}{6}.\]
Calculating the two solutions:
\[x_1 = \frac{-4 + 8}{6} = \frac{4}{6} = \frac{2}{3}, \quad x_2 = \frac{-4 - 8}{6} = \frac{-12}{6} = -2.\]
The zeros of the polynomial are \(-2, 1/3, 2/3\).
Examples
In engineering and physics, solving for the zeros of a polynomial like this is crucial for understanding the stability of systems. For example, if this polynomial represented the characteristic equation of a mechanical system, finding its zeros would allow engineers to determine the system's natural frequencies and identify potential resonances. By identifying all zeros, we gain insight into the system's behavior under different conditions, helping us ensure safety and optimal performance in real-world designs.