Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=x^{3}-3 x^{2}-4 x+12\)

Step-by-Step Solution

Answer

The real solutions of the equation \(f(x)=x^{3}-3 x^{2}-4 x+12=0\) are \(x= -2, x= 1, x=6\).

Step 1: Rational Root Theorem

Apply the Rational Root Theorem to suggest all possible roots. The Rational Root Theorem suggests that any possible rational root, p/q, of the equation \(ax^{n} + bx^{n-1} + ... + z = 0\) would have p as a factor of the constant term (the 'z' term) and q as a factor of the leading coefficient (the coefficient 'a' on the term of highest power). For this equation, all factors of 12 (±1, ±2, ±3, ±4, ±6 and ±12) could be potential roots.

Step 2: Testing Potential Roots

Test these potential roots until one is found that makes \(f(x)=0\). Using synthetic division or direct substitution, it is found that -2, 1, and 6 are roots of the equation.

Step 3: Writing the Final Solution

The real values of \(x\) in the given function \(x^{3}-3 x^{2}-4 x+12\) that make \(f(x)=0\) are -2, 1, and 6.

Key Concepts

Rational Root Theorem

Understanding the Rational Root Theorem is crucial when it comes to solving polynomial equations. This theorem provides a systematic way to list all possible rational roots of a polynomial equation of the form
\(ax^{n} + bx^{n-1} + ... + z = 0\).
Here, if we have a potential rational root in the form \(\frac{p}{q}\), then 'p' must be a factor of the constant term 'z', and 'q' must be a factor of the leading coefficient 'a'. In the exercise provided, the leading coefficient is 1 (implied before \(x^{3}\)) and the constant term is 12. Thus, the potential roots are all the factors of 12, divided by the factors of 1. This yields a list of possible rational roots: ±1, ±2, ±3, ±4, ±6, and ±12.
The beauty of the Rational Root Theorem lies in its simplicity and effectiveness in narrowing down the candidates for potential roots, which are otherwise infinite. By focusing only on these possibilities, students can save time and effort when solving polynomial equations.

Synthetic Division

Synthetic division is a shortcut method that simplifies the process of dividing a polynomial by a binomial of the form \((x - c)\), where 'c' is a constant. It is particularly handy when testing possible roots identified by the Rational Root Theorem.
In the exercise, synthetic division can be used to quickly test if -2, 1, or 6 are roots of the polynomial \(x^{3}-3x^{2}-4x+12\). Here's how it works:

• Write down the coefficients of the polynomial.
• Bring down the leading coefficient.
• Multiply it by the potential root and write the result under the next coefficient.
• Add downwards and repeat the multiply-add process.

If the final value (the remainder) is zero, then the tested value is a root. This approach is not only less cumbersome than long division but also lends itself nicely to quick calculations, making it invaluable for students aiming for efficiency.

Polynomial Roots

Polynomial roots are the values for which the polynomial equation equals zero. In simpler terms, they are the solutions to the equation \(f(x) = 0\). These roots can be real or complex numbers. Real roots are of special interest as they can often be graphed on a standard Cartesian plane and represent the points where the graph of the polynomial intersects the x-axis.
The exercise aims to find all real values of \(x\) that make \(f(x) = 0\) for the polynomial \(f(x) = x^{3}-3x^{2}-4x+12\). Using the Rational Root Theorem to identify possible roots and synthetic division to test them, students can determine the real roots efficiently. In this particular case, -2, 1, and 6 are found to be the real roots, which suggests the graph of \(f(x)\) crosses the x-axis at these points. Understanding the concept of roots is essential not just for solving equations, but also for comprehending the graphical behaviour of polynomials.