Problem 60 - College Algebra with Corequisite Support

Question

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=4 x^{5}-10 x^{4}+8 x^{3}+x^{2}-8 $$

Step-by-Step Solution

Answer

Possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4} \).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem states that any rational root, expressed in its simplest form as \( \frac{p}{q} \), of a polynomial equation with integer coefficients is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. For the polynomial \( f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8 \), the constant term is \(-8\) and the leading coefficient is \(4\).

Step 2: Find the Factors of the Constant Term

The constant term of the polynomial is \(-8\). The factors of \(-8\) can be \(\pm 1, \pm 2, \pm 4, \pm 8\).

Step 3: Find the Factors of the Leading Coefficient

The leading coefficient is \(4\). The factors of \(4\) are \(\pm 1, \pm 2, \pm 4\).

Step 4: Construct Possible Rational Zeros

Using the Rational Root Theorem, list the possible rational zeros as \( \frac{p}{q} \), where \( p \) is a factor of \(-8\) and \( q \) is a factor of \(4\). The potential rational zeros are:\[ \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{7}{2}, \pm \frac{7}{4} \]

Step 5: Simplify the List of Rational Zeros

Upon review for duplicate values, simplify the list of possible rational zeros. The simplified list is:\[ \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4} \]

Step 6: Verify Potential Rational Zeros

The exercise asks us only to list them, not verify them, but in practical circumstances, you would substitute these values into the polynomial to check whether they are actual zeros.

Key Concepts

Polynomial functions

Polynomials are mathematical expressions consisting of variables raised to various powers, with coefficients that are integers. They can be likened to versatile building blocks in algebra, defining a wide range of shapes and paths in a graph. In general, a polynomial function is expressed in the form:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]Here, \(a_n\), \(a_{n-1}\), the coefficients, determine the weight of each term, while \(a_0\) is the constant term. The degree of the polynomial is determined by the highest power of \(x\) present, marking the complexity and number of potential roots or zeros the polynomial might have.

• Higher degree polynomials can have more turning points and zero crossings.
• Their graph can twist and turn, crossing the x-axis at each rational or irrational zero.

Understanding polynomial functions is crucial in solving equations and analyzing and predicting trends by modeling real-world situations within mathematics.

Factors of integers

To understand factors of integers, think of breaking down a number into smaller whole numbers which, when multiplied, give back the original number. These smaller numbers are known as the factors. For example, the factors of the integer 8 include 1, 2, 4, and 8 itself. To find the factors of a number, list all pairs of numbers whose product equals the original integer. Factors are always integer numbers, and they include both positive and negative values because multiplying two negative numbers yields a positive product.

• The number 1 and the number itself are always factors of any given integer.
• Factors are used in the Rational Root Theorem to determine potential rational zeros of polynomial functions.

Recognizing factors is a key skill in simplifying equations and solving polynomial problems efficiently.

Rational zeros

Rational zeros of a polynomial are those solutions or roots that can be expressed in the form of a simple fraction \(\frac{p}{q}\). The presence of rational zeros offers significant insights into the behavior of a polynomial function.The Rational Root Theorem aids in finding these potential zeros by examining the

• factors of the constant term, \(p\).
• factors of the leading coefficient, \(q\).

By doing so, we construct all possible fractions \(\frac{p}{q}\) that could potentially be zeros.
The process involves evaluating each possible zero to see if it satisfies the polynomial equation, making it truly a zero of the polynomial.This method provides a practical approach for zeroing in on accurate solutions without grappling with all real and complex numbers, streamlining the search for zeros in polynomial functions.