- Start by factoring out \(x\) to get \(f(x) = x(2x^3 - x^2 - 18x + 9)\).
- Use grouping on the remaining cubic expression to factor it as \((2x - 1)(x^2 - 9)\).
- Further decompose the difference of squares \((x^2 - 9)\) into \((x - 3)(x + 3)\) to reach the final factored form: \(f(x) = x(2x - 1)(x - 3)(x + 3)\).
- Determine the four zeros by setting each factor to zero, resulting in \(\boxed{-3, 0, 1/2, 3}\).
Explanation
- Initial Analysis and Factoring out x
To find the zeros of the function \(f(x) = 2x^4 - x^3 - 18x^2 + 9x\), we need to solve the equation \(f(x) = 0\). This is a fourth-degree polynomial equation. Our first step is to identify any common factors. We can see that every term in the expression has at least one power of \(x\), so we can factor out \(x\):
\[f(x) = x(2x^3 - x^2 - 18x + 9).\]
Now we need to focus on the cubic polynomial \(2x^3 - x^2 - 18x + 9 = 0\).
- Factoring by Grouping
Next, we use the method of grouping to factor the cubic polynomial. We group the first two terms and the last two terms:
\[(2x^3 - x^2) - (18x - 9).\]
Now, we factor out the greatest common factor from each group:
- From \(2x^3 - x^2\), we factor out \(x^2\), giving \(x^2(2x - 1)\).
- From \(18x - 9\), we factor out \(9\), giving \(9(2x - 1)\).
Substituting these back, we get:
\[x^2(2x - 1) - 9(2x - 1).\]
We can now factor out the common binomial factor \((2x - 1)\):
\[(2x - 1)(x^2 - 9).\]
Combining this with the factor \(x\) from the first step, the fully factored form is \(f(x) = x(2x - 1)(x^2 - 9)\).
- Identifying the Zeros
The term \((x^2 - 9)\) is a difference of squares, which follows the pattern \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 3\), so \((x^2 - 9) = (x - 3)(x + 3)\).
Substituting this back into our expression gives:
\[f(x) = x(2x - 1)(x - 3)(x + 3).\]
To find the zeros, we set each factor equal to zero:
- \(x = 0\)
- \(2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}\)
- \(x - 3 = 0 \implies x = 3\)
- \(x + 3 = 0 \implies x = -3\)
The roots of the function are \(-3, 0, \frac{1}{2},\) and \(3\). Ordered from left to right (least to greatest), they are \(-3, 0, \frac{1}{2},\) and \(3\).
Examples
Polynomial factorization is widely used in physics and engineering to analyze system stability and resonance. For example, when studying the motion of a mechanical system or the frequency response of an electrical circuit, the characteristic equation is often a polynomial. Finding the zeros (roots) of these equations allows engineers to determine the points at which the system equilibrium changes or oscillates, ensuring that designs are robust and operate correctly under varying conditions.