Problem 431 - Precalculus

Question

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 Leading Coefficient and the Constant Term

For the polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), identify the leading coefficient and the constant term. The leading coefficient is 2, and the constant term is 5.

Step 2: List Factors of the Constant Term

List all factors of the constant term, which is 5. The factors of 5 are \( \pm 1, \pm 5 \).

Step 3: List Factors of the Leading Coefficient

List all factors of the leading coefficient, which is 2. The factors of 2 are \( \pm 1, \pm 2 \).

Step 4: Use the Rational Root Theorem

According to the Rational Root Theorem, possible rational zeros of the polynomial are given by \( \frac{\text{factors of constant term}}{\text{factors of leading coefficient}} \).

Step 5: List Possible Rational Zeros

Combine the factors from Steps 2 and 3 to list all possible rational zeros. These are:\[\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}\]

Key Concepts

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. It is an important part of understanding the polynomial's behavior and is a key factor when applying the Rational Root Theorem. In our example polynomial, \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the term with the highest degree is \( 2x^3 \).
Therefore, the leading coefficient is 2.Identifying the leading coefficient is crucial because it helps in determining the possible rational roots. By knowing that the leading coefficient is 2, we understand that it influences the divisor pool when we apply the Rational Root Theorem. This coefficient tells us about the spread of possible values for the zeros and can have implications on the polynomial's end behavior if evaluated with graphical methods.

Constant Term

The constant term in a polynomial is the term that does not contain a variable. It is the term you would get if you substituted 0 for all the variables in the polynomial. In our polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the constant term is 5.The constant term plays a significant role in determining the possible rational zeros of a polynomial. When applying the Rational Root Theorem, the factors of the constant term contribute to the numerator of the fractions representing potential zeros. In simple terms, the constant term helps us explore all possible whole number zeros the polynomial might have. If the constant term were zero, the calculations and possible rational roots would change dramatically.

Rational Root Theorem

The Rational Root Theorem is a mathematical principle used to find all potential rational zeros of a polynomial. It states that any possible rational solution \( \frac{p}{q} \), where \( p \) and \( q \) are integers, must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient of the polynomial. In the context of our example polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), this theorem helps us list potential rational roots:
- Factors of the constant term 5 are \( \pm 1, \pm 5 \).- Factors of the leading coefficient 2 are \( \pm 1, \pm 2 \).Using the Rational Root Theorem, we generate possible rational roots by dividing each factor of the constant term by each factor of the leading coefficient. This gives us a comprehensive list of potential zeros without needing to resort to computational guessing.

Factors of Polynomials

Factors of a polynomial are expressions that can be multiplied together to obtain the polynomial itself. Finding these factors is a fundamental aspect of solving polynomial equations.In the process of determining the rational zeros of a polynomial, understanding its factors is critical. By identifying the factors of both the leading coefficient and the constant term, we can list possible rational zeros, which may later be verified through substitution or synthetic division. For example, in the polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), knowing that 2 and 5 are the leading coefficient and constant term, respectively, guides us in listing the factors:

• Factors of 2: \( \pm 1, \pm 2 \)
• Factors of 5: \( \pm 1, \pm 5 \)

Identifying these factors allows us to properly utilize the Rational Root Theorem, giving us potential rational zeros such as \( \pm 1 \), \( \pm \frac{1}{2} \), \( \pm 5 \), and \( \pm \frac{5}{2} \). These expressions represent combinations of our identified factors, each one playing a vital role under the theorem's guidance.