Problem 3 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{3}-8 x-3 $$

Step-by-Step Solution

Answer

The only rational zero of the polynomial is \(x = 3\).

Step 1: Identify Possible Rational Zeros

To find the rational zeros, we first use the Rational Root Theorem. This theorem states that any possible rational zero, \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. In the polynomial \(f(x) = x^3 - 8x - 3\), the constant term is \(-3\) and the leading coefficient is \(1\). The potential values for \(p\) are \(\pm 1, \pm 3\), and for \(q\), these are simply \(\pm 1\). Therefore, the possible rational zeros are \(\pm 1, \pm 3\).

Step 2: Check Each Possibility by Substitution

Next, we test each of these possible rational zeros by substituting them into the polynomial \(f(x)\).- Substitute \(x = 1\): \(f(1) = 1^3 - 8(1) - 3 = 1 - 8 - 3 = -10\) (not zero)- Substitute \(x = -1\): \(f(-1) = (-1)^3 - 8(-1) - 3 = -1 + 8 - 3 = 4\) (not zero)- Substitute \(x = 3\): \(f(3) = 3^3 - 8(3) - 3 = 27 - 24 - 3 = 0\) (zero found)- Substitute \(x = -3\): \(f(-3) = (-3)^3 - 8(-3) - 3 = -27 + 24 - 3 = -6\) (not zero)

Step 3: Verify Found Zero

Since \(x = 3\) is a zero, we need to verify by division or check if there are any further rational zeros. To ensure \(x = 3\) is a zero, we can use synthetic division to divide \(f(x)\) by \(x - 3\), confirming it yields a remainder of 0. This confirms \(x = 3\) is a valid zero.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables with coefficients. These functions take different forms, such as linear, quadratic, cubic, and so on, based on the highest power of the variable they contain. The general form of a polynomial function in one variable is given by:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients.
• \(n\) is a non-negative integer representing the degree of the polynomial.
• The highest exponent of the variable is the degree of the polynomial, which determines the number of roots (or zeros) the polynomial can have.

Polynomial functions are continuous and smooth over their domain, making them useful for modeling a wide range of real-world situations. In this specific exercise, the polynomial function is a cubic polynomial, \(f(x) = x^3 - 8x - 3\), which means it can have up to 3 real zeros.

Rational Zeros

The Rational Root Theorem is a vital tool in finding rational zeros of polynomial functions. It provides a way to list all possible rational numbers that could be zeros of the polynomial based on the coefficients of the terms. Here's how it works:

• If a polynomial has a rational zero \(\frac{p}{q}\), then \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
• This gives us a finite set of possible rational zeros that we can test in the polynomial function.

In the exercise, the constant term of the polynomial \(f(x) = x^3 - 8x - 3\) is \(-3\), and the leading coefficient is \(1\). Hence, the possible rational zeros are the factors of \(-3\) divided by the factors of \(1\), which are \(\pm 1, \pm 3\). By testing these potential zeros through substitution into the polynomial, we find that \(x = 3\) is indeed a zero of the polynomial.

Synthetic Division

Once a potential zero is found for a polynomial function, synthetic division is an efficient method to verify this zero and simplify the polynomial for finding further zeros. Synthetic division is performed using the following steps:

• Write down the coefficients of the polynomial in descending order of powers.
• The potential zero, say \(c\), is placed outside the division symbol.
• Bring down the leading coefficient to the bottom row.
• Multiply this leading coefficient by \(c\) and write the result under the next coefficient. Then add them together and continue this process across all coefficients.
• If the last number you compute (the remainder) is zero, \(c\) is indeed a zero of the polynomial.

For the exercise's polynomial \(f(x) = x^3 - 8x - 3\), after testing and finding \(x = 3\) as a zero, synthetic division confirms it leaves a remainder of zero, solidifying it as a root of the equation. This reduces the complexity of the polynomial and aids in finding additional zeros, if they exist.