Question

Find all real zeros of the function. $$f(x)=x^3-4 x^2-x+4$$

Short answer

The real roots of the function \(f(x) = x^3-4x^2-x+4\) are \(x=1\), \(x=2+\sqrt{5}\), and \(x=2-\sqrt{5}\).

Step-by-step solution

Set the function equal to zero

We start by setting the function equal to zero: \(x^3 - 4x^2 - x + 4 = 0\).

Factor the equation

Next, we factor the equation. Notice that there is a pattern that can be factored out as follows: \((x-1)(x^2-4x-4)=0\).

Set each factor equal to zero

Now we have our two factors, we can set each equal to zero in order to solve for x: \n1. \(x - 1 = 0\), solving this equation gives: \(x=1\).\n2. \(x^2 - 4x - 4 = 0\). This is a quadratic equation which can be solved using the quadratic formula \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4*1*(-4)}}{2*1}\) which yields \(x=2\pm\sqrt{5}\).

Find the real roots

From the above we get three real roots: \(x=1\), \(x=2+\sqrt{5}\), and \(x=2-\sqrt{5}\).

Key concepts

Factoring Polynomials

Understanding how to factor polynomials is crucial when it comes to finding real zeros of a function. Factoring involves expressing a polynomial as the product of its simpler, non-zero factors. This is often the first step when attempting to solve polynomial equations, since factors set to zero can reveal the roots of the original polynomial.

For instance, in the given exercise, the polynomial is factored into \(x-1)\) and \(x^2-4x-4\). A key strategy for factoring involves looking for patterns such as the difference of squares, grouping, or using the distributive property to find common factors. It's important to practice recognizing these patterns to factor efficiently.

Once the polynomial is factored, the real zeros are easier to find because each factor set to zero allows you to solve for possible values of \(x\). Consequently, mastering polynomial factoring techniques can vastly improve problem-solving skills in algebra.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are polynomials of the second degree, typically in the form of \(ax^2 + bx + c = 0\). The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) provides the solutions to these equations by accounting for all possible scenarios and is derived from the process of completing the square.

The exercise solution applies the quadratic formula to the factor \(x^2 - 4x - 4\). The real zeros, or roots, can be identified once \(a\), \(b\), and \(c\) are plugged into the formula, yielding \(2 \pm \sqrt{5}\) when simplified. Students must pay special attention to the discriminant—\(b^2 - 4ac\)—as it indicates the number and type of solutions. A positive discriminant results in two distinct real roots, which was the case in our exercise. A zero discriminant means one real root, and a negative discriminant leads to complex roots. Knowing the quadratic formula is essential as it is a reliable method to find real zeros of second-degree polynomials.

Roots of Polynomial Functions

The roots of polynomial functions are the values of \(x\) that make the polynomial equal to zero. These roots are also referred to as zeros, x-intercepts, or solutions to the polynomial equation. In the context of our exercise, finding the real zeros is the ultimate goal, and achieving this requires setting the polynomial \(f(x)\) to zero and solving for \(x\).

Every polynomial function of degree \(n\) has exactly \(n\) complex roots, some of which may be real. The Fundamental Theorem of Algebra guarantees at least one complex root, but the challenge is often determining the real ones among them. Factoring the polynomial into linear and/or irreducible quadratic factors, as shown in the solution, paves the way to finding real zeros.

Real roots have significant graphical implications since they represent the points where the function's graph crosses the x-axis. The exercise clearly exemplifies that after factoring and simplifying, the polynomial's real zeros are \(1\), \(2+\sqrt{5}\), and \(2-\sqrt{5}\). These values are not just solutions to an equation but also key to analyzing the shape and behavior of the polynomial's graph.