Problem 8 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The real zeros of the function \(g(x)=x^{3}-4 x^{2}-x+4\) are \(x=1\) and \(x=-2\).

Step 1: Set up the Equation

The first step is to set up our function equal to zero. This gives us the equation \(x^{3}-4 x^{2}-x+4 = 0\). This is what we are solving.

Step 2: Derby's Rational Roots Theorem

Derby's Rational Roots Theorem states that if a rational number expressed in lowest terms as \(p/q\) is a root of a polynomial equation with integer coefficients, then \(p\) must be a factor of the constant term (in this case \(4\)) and \(q\) must be a factor of the coefficient of the highest degree term (in this case \(1\)). Thus for this polynomial, the possible rational roots are \(\pm1, \pm2, \pm4\). Now substitute each of these roots into the polynomial equation, and see which substitutions produce \(0\).

Step 3: Identify the Real Roots

After substitution, it can be checked that the real roots are \(x=1\) and \(x=-2\) by substituting these values into the original equation, which simplifies to zero, thereby demonstrating that these are indeed the roots.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool used to find potential rational roots of a polynomial equation. It applies when a polynomial has integer coefficients. In its essence, this theorem states that for a polynomial like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\), any potential rational root, expressed in its simplest form as \(\frac{p}{q}\), must meet two criteria:

• \(p\), the numerator, has to be a factor of the constant term, \(a_0\).
• \(q\), the denominator, must be a factor of the leading coefficient, \(a_n\).

So, how does this help us? By limiting the number of potential rational roots, we focus our efforts on testing these likely candidates instead of guessing blindly. For example, in the function \(g(x) = x^3 - 4x^2 - x + 4\):

• Here, the constant term \(a_0\) is 4, and its factors are \(\pm1, \pm2, \pm4\).
• The leading coefficient is 1, which only has \(\pm1\) as its factors.

Thus, the potential rational roots according to the Rational Root Theorem are \(\pm1, \pm2, \pm4\). Testing these values will reveal which are truly zeroes of the polynomial function.

Real Roots

Real roots of polynomial equations are the solutions where the polynomial equals zero and are also real numbers, not involving imaginary or complex numbers. Identifying these real roots is vital as they indicate the points where a polynomial intersects the x-axis on a graph. In the case of the polynomial \(x^3 - 4x^2 - x + 4\), analyzing the rational root candidates \(\pm1, \pm2, \pm4\) leads us to identify \(x=1\) and \(x=-2\) as real roots.
How do we confirm whether a candidate is a real root? We substitute the value into the polynomial equation and simplify. If the equation yields zero, then it's a real root. For \(x=1\):

• Substituting gives \(1^3 - 4(1)^2 - 1 + 4 = 0\).
• The expression simplifies to zero, proving \(x=1\) is a real root.

Similarly, for \(x=-2\):

• Substituting yields \((-2)^3 - 4(-2)^2 - (-2) + 4 = 0\).
• This also simplifies to zero, confirming \(x=-2\) as a real root.

Real roots are especially significant because they are the 'visible' solutions when graphing, unlike complex roots.

Polynomial Equation Solutions

Solving polynomial equations involves finding all the possible roots or solutions, which can include real roots, imaginary roots, or complex roots, depending on the degree of the polynomial equation and the nature of its coefficients. The degree of the polynomial equation, seen in the highest power of the variable, determines the number of possible solutions. For example, our function \(x^3 - 4x^2 - x + 4 = 0\) is a third-degree polynomial, suggesting that it can have up to three roots.
To solve such polynomials:

• Start with the Rational Root Theorem to predict likely rational roots.
• Find the real roots by substituting these rational roots into the polynomial.
• Once real roots are identified, if unknowns remain or need verification, use methods like polynomial division, synthetic division, or factorization for further solving.

Hence, the process involves narrowing down potential roots, verifying them, and then extrapolating to possibly find remaining complex or repeated roots. It’s a strategic blend of equations and tested trial, ensuring we capture all the potential solutions to the polynomial equation.