Problem 57 - College Algebra

Question

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

Step-by-Step Solution

Answer

Possible rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2} \).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem states that if a polynomial has a rational zero \( \frac{p}{q} \), where \( p \) and \( q \) are integers, then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

Step 2: Determine Factors of Constant and Leading Coefficient

In the polynomial \( f(x)=2x^{3}+3x^{2}-8x+5 \), the constant term is 5 and the leading coefficient is 2. The factors of 5 are \( \pm 1, \pm 5 \). The factors of 2 are \( \pm 1, \pm 2 \).

Step 3: List All Possible Rational Zeros

According to the Rational Root Theorem, the possible rational zeros are \( \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{5}{1}, \pm \frac{5}{2} \). Simplifying these fractions, the possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2} \).

Key Concepts

Polynomial Zeros

Polynomials are equations that can have several terms, each consisting of a coefficient, a variable raised to a power, and sometimes a constant. The zeros of a polynomial are the values of the variable that make the whole polynomial equal to zero. Finding these zeros is often essential in mathematics because they are the solutions or roots of the polynomial. For example, if you have a polynomial function represented by \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), its zeros are the values of \( x \) for which \( f(x) = 0 \). Understanding polynomial zeros could also help in graphing the polynomial function, as they show where the polynomial intersects the x-axis.

• Zeros can be real or complex numbers.
• Methods to find zeros include factoring, synthetic division, and using the Rational Root Theorem.

Factors of Constant Term

The constant term in a polynomial is what does not change, regardless of the variable's value. In the polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the constant term is 5. To use the Rational Root Theorem effectively, identifying the factors of this constant term is crucial. These factors are numbers that multiply to give the constant term. Some key points about factors of constant terms:

• Factors of 5 are \( \pm 1, \pm 5 \).
• These factors help in determining possible numerators for rational zeros.
• Understanding these factors helps set the stage for limiting potential rational solutions.

Leading Coefficient Factors

The leading coefficient in a polynomial is the coefficient of the term with the highest exponent, critical in the Rational Root Theorem. For \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the leading term is \( 2x^3 \), thus making the leading coefficient 2. To find potential rational zeros, you should consider the factors of this leading coefficient. The importance of leading coefficient factors includes:

• Factors of 2 are \( \pm 1, \pm 2 \).
• These factors comprise the denominators in rational zero candidates.
• Recognizing them aids in narrowing down possible rational zeros efficiently.

Rational Zeros

Rational zeros of a polynomial are zeros that can be expressed as fractions of integers, making them essential in analyzing these mathematical functions. According to the Rational Root Theorem, each potential rational zero can be expressed in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Using the polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), we determine:

• Possible numerators from factors of 5: \( \pm 1, \pm 5 \).
• Possible denominators from factors of 2: \( \pm 1, \pm 2 \).
• Listing all combinations, we derive potential rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2} \).

By investigating these rational numbers, you can identify which ones truly zero the polynomial, helping solve polynomial equations.