Problem 14 - Algebra 2

Question

List all of the possible rational zeros of each function. \(p(x)=3 x^{3}-5 x^{2}-11 x+3\)

Step-by-Step Solution

Answer

The possible rational zeros are \( \pm 1, \pm 3, \pm \frac{1}{3} \).

Step 1: Define the Rational Root Theorem

The Rational Root Theorem states that for any rational solution \( \frac{p}{q} \) of a polynomial equation with integer coefficients, \( p \) (a factor of the constant term) and \( q \) (a factor of the leading coefficient) must both be integers.

Step 2: Identify the Constant and Leading Coefficients

Identify the constant term and leading coefficient of the polynomial. For \( p(x) = 3x^3 - 5x^2 - 11x + 3 \), the constant term is 3, and the leading coefficient is also 3.

Step 3: List the Factors of the Constant Term

The factors of the constant term (3) are \( \pm 1, \pm 3 \).

Step 4: List the Factors of the Leading Coefficient

The factors of the leading coefficient (3) are \( \pm 1, \pm 3 \).

Step 5: Combine Factors to Form Potential Rational Zeros

Based on the Rational Root Theorem, any rational zero will be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. The possible rational zeros are:\[ \pm 1, \pm 3, \pm \frac{1}{3}, \pm \frac{3}{3} \] Since \( \frac{3}{3} \) simplifies to 1, this list can be reduced to \( \pm 1, \pm 3, \pm \frac{1}{3} \).

Key Concepts

Polynomial Equation

A polynomial equation is a mathematical expression that sets a polynomial equal to zero. It generally takes the form of \( p(x) = 0 \). The main component of a polynomial equation is the polynomial itself, usually expressed as a sum of terms.

• Each term consists of a coefficient multiplied by a variable raised to a whole number power.
• The degree of the polynomial is the highest power of the variable present in the equation.
• In the given exercise, the polynomial \( p(x) = 3x^3 - 5x^2 - 11x + 3 \) is a cubic polynomial because the highest exponent on the variable is 3.

Understanding polynomial equations is important in algebra and calculus because they frequently model real-world scenarios, from physics to finance. Solving polynomial equations involves finding the 'roots' of the polynomial, where the function equals zero.

Integer Coefficients

Integer coefficients are numbers without fractional or decimal parts that multiply the variables in a polynomial. In a polynomial equation, the coefficients could be positive, negative, or zero, but they are always whole numbers.

• For the polynomial \( p(x) = 3x^3 - 5x^2 - 11x + 3 \), the coefficients are 3, -5, -11, and 3, all of which are integers.
• Integer coefficients greatly simplify calculations and analysis of polynomials.
• They play a critical role in the Rational Root Theorem, which applies only to polynomials with integer coefficients.

The Rational Root Theorem, a key concept in algebra, cannot be used on polynomials with non-integer coefficients. Therefore, verifying that a polynomial has integer coefficients is an essential step before applying the theorem.

Rational Zeros

Rational zeros, also known as rational roots, are rational numbers that satisfy the polynomial equation when substituted for the variable. According to the Rational Root Theorem, these zeros, when they exist, have specific properties with respect to the polynomial's integer coefficients.

• The possible rational zeros of a polynomial like \( p(x) = 3x^3 - 5x^2 - 11x + 3 \) are determined by the factors of the constant term and the leading coefficient.
• In our example, these factors are \( \pm 1, \pm 3 \) for both the constant term and leading coefficient, providing the potential rational zeros: \( \pm 1, \pm 3, \pm \frac{1}{3} \).
• The theorem simplifies the search for rational solutions, giving a finite set of candidates rather than having to test infinitely many numbers.

Finding rational zeros can dramatically simplify solving polynomial equations, as each rational zero corresponds to a factor of the polynomial in linear form, aiding in polynomial division and further simplification.