Problem 17 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=2.5 x^{3}+x^{2}+0.6 x+0.1 $$

Step-by-Step Solution

Answer

No rational zeros; none of the possible roots satisfy \( f(x) = 0 \).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem states that any rational root \( \frac{p}{q} \) of the polynomial \( f(x) \) is such that \( p \) divides the constant term and \( q \) divides the leading coefficient. For \( f(x) = 2.5x^3 + x^2 + 0.6x + 0.1 \), the constant term is 0.1 and the leading coefficient is 2.5.

Step 2: Determine Possible Values of \( p \) and \( q \)

\( p \) must be a factor of 0.1 and \( q \) must be a factor of 2.5. The factors of 0.1 are \( \pm 0.1, \pm 0.01 \), and the factors of 2.5 are \( \pm 2.5, \pm 1, \pm 0.5 \).

Step 3: List Possible Rational Roots

Using the factors of \( p \) and \( q \), the possible rational roots are the fractions \( \frac{p}{q} \). Thus, the possible rational roots are \( \pm 0.04, \pm 0.02, \pm 0.4, \pm 0.2, \pm 0.1, \pm 0.05 \).

Step 4: Evaluate Roots Using Synthetic Division

Test each possible root by substituting them into the polynomial or using synthetic division to see if they yield zero. For simplicity, let's evaluate a couple using direct substitution.

Step 5: Test \( x = 0.1 \)

Substitute \( x = 0.1 \) in \( f(x) \):\[ f(0.1) = 2.5(0.1)^3 + (0.1)^2 + 0.6(0.1) + 0.1 = 0.0025 + 0.01 + 0.06 + 0.1 = 0.1725 \]Since \( f(0.1) eq 0 \), \( x = 0.1 \) is not a root.

Step 6: Test \( x = -0.1 \)

Substitute \( x = -0.1 \) in \( f(x) \):\[ f(-0.1) = 2.5(-0.1)^3 + (-0.1)^2 + 0.6(-0.1) + 0.1 = -0.0025 + 0.01 - 0.06 + 0.1 = 0.0475 \]Since \( f(-0.1) eq 0 \), \( x = -0.1 \) is not a root.

Step 7: Conclude on the Rational Roots

After testing a reasonable number of possible roots, it seems none of the suggested rational roots satisfy \( f(x) = 0 \). This implies that \( f(x) \) likely has no rational roots.

Key Concepts

Polynomial Roots

Finding the roots of a polynomial involves determining the values of the variable that make the polynomial equal to zero. These values are the `'roots'` or `'zeros'` of the polynomial. For a polynomial like \( f(x) = 2.5x^3 + x^2 + 0.6x + 0.1 \), we are interested in the values of \( x \) that satisfy \( f(x) = 0 \).
To find these roots, especially rational roots, methods like the Rational Root Theorem are often used. This theorem provides a systematic approach to finding potential rational roots by looking at the factors of the constant term and the leading coefficient.
Rational roots are those roots that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers. In our case, potential roots are thoroughly explored by this approach, leading us to list and test potential candidates.

Synthetic Division

Synthetic division is a compact and efficient method of dividing polynomials, especially useful in checking if a specific value is a root of a polynomial. Compared to long division, synthetic division is faster and involves fewer steps.
The method involves writing down the coefficients of the polynomial and then performing operations based on a hypothesized root. If the result of synthetic division leaves us with a remainder of zero, then the value tested is indeed a root of the polynomial.
For the polynomial \( f(x) = 2.5x^3 + x^2 + 0.6x + 0.1 \), synthetic division helps quickly evaluate the likeliness of a candidate being a root by processing through simple arithmetic steps without dividing the full polynomial explicitly. It is a trial-and-error process based on possible rational roots provided by the Rational Root Theorem.

Possible Rational Roots

The concept of possible rational roots is derived from the Rational Root Theorem, which limits the number of rational numbers you need to test when looking for the roots of a polynomial. According to the theorem, any rational root, expressed as \( \frac{p}{q} \), of a polynomial will occur such that \( p \) divides the constant term, and \( q \) divides the leading coefficient.
To determine these roots for \( f(x) = 2.5x^3 + x^2 + 0.6x + 0.1 \), we evaluate the factors:

• Factors of constant term 0.1: \( \pm 0.1, \pm 0.01 \)
• Factors of leading coefficient 2.5: \( \pm 2.5, \pm 1, \pm 0.5 \)

By pairing these factors appropriately, we create a possible list of rational roots. The list might include values such as \( \pm 0.04, \pm 0.02, \pm 0.4, \pm 0.2, \pm 0.1, \pm 0.05 \). It's important to evaluate whether these potential roots satisfy \( f(x) = 0 \) through substitution or synthetic division.