Question

Find all real zeros of the function. $$f(x)=3 x^{3}-19 x^{2}+33 x-9$$

Short answer

The real zeros of the function \(f(x) = 3x^{3} - 19x^{2} + 33x - 9\) are \(x = 1\) and \(x = 3\), with 3 being a repeated root.

Step-by-step solution

Initial Observation

Firstly, notice the equation: \(f(x)=3x^{3} - 19x^{2} + 33x - 9\).

Setting the Function Equal to Zero

The main task is to find the real zeros, so set the function equal to zero. That leads to the following equation: \(3x^{3} - 19x^{2} + 33x - 9 = 0\).

Finding Possible Rational Roots

Using the rational root theorem, which states that if a polynomial in one variable has a rational root, then that root can be expressed as the ratio of a factor of the constant term to a factor of the leading coefficient, list all the possible rational roots. In this case, it will include factors of 9 over factors of 3. The possibilities include \(\pm 1, \pm 3, \pm 9\).

Testing for Rational Roots

By trying these possible roots into the equation, one finds that 1 and 3 are indeed roots of the polynomial. That means, when x equals to 1 or 3, the function is equal to 0.

Factoring the polynomial

Knowing some roots of the polynomial allows to factor the polynomial further. It turns out that the function can be factored into \(f(x) = 3(x - 1)(x - 3)^{2}\). This indicates that the real zeros of the equation are \(x = 1\) and \(x = 3\) with 3 being a repeated root.

Key concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding the potential rational solutions to polynomial equations. Specifically, it states that any possible rational root of a polynomial equation in the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 \) can be determined by taking the factors of the constant term \( a_0 \) divided by the factors of the leading coefficient \( a_n \).
For example, consider the polynomial \( f(x) = 3x^3 - 19x^2 + 33x - 9 \). Here, the constant term is \(-9\) and the leading coefficient is \(3\). Therefore, the possible rational roots could be \( \pm 1, \pm 3, \pm 9 \) divided by \( \pm 1, \pm 3 \).
This yields possible rational roots of \( \pm 1, \pm 3, \pm 9 \). It is important to note that not all potential roots will necessarily be actual solutions. Therefore, these candidates must be tested in the polynomial to identify actual roots.

Factoring Polynomials

Factoring is a method of expressing a polynomial as the product of its simpler polynomials. Once the rational roots are identified, you can begin to factor the polynomial accordingly. Factoring simplifies solving for zeros, as it breaks the polynomial into manageable parts.
Consider our polynomial \( f(x) = 3x^3 - 19x^2 + 33x - 9 \), which can be factored once some roots, such as 1 and 3, are identified. In this case, factor by taking \( (x-1) \) and \( (x-3)^2 \) to get \( 3(x - 1)(x - 3)^2 \).
This factoring reveals that • \( x = 1 \) is a straightforward root, and • \( x = 3 \) is a repeated (or double) root, as indicated by the \((x - 3)^2\). The multiplication by \( 3 \) ensures that the original polynomials' structure is maintained.

Real Zeros of a Function

Finding the real zeros of a function involves determining the \( x \)-values at which the function equals zero. These zeros are significant for understanding the behavior and graph of a function as they represent the x-intercepts.
When dealing with the polynomial \( f(x) = 3x^3 - 19x^2 + 33x - 9 \), the steps for finding real zeros involve employing the Rational Root Theorem to test potential rational roots. After testing, it is discovered that \( x = 1 \) and \( x = 3 \) are indeed real zeros, with \( x = 3 \) being a repeated zero. The factored form \( 3(x-1)(x-3)^2 \) clearly shows this.
Understanding these zeros is paramount as • They mark points where the polynomial intersects the x-axis on a graph. • They can help in sketching or interpreting the behavior of the graph near those points, especially if certain zeros are repeated.