Problem 91 - Algebra 2

Question

List all possible rational roots for each equation. Then use the Rational Root Theorem to find each root. $$ 2 x^{3}-11 x^{2}-x+30=0 $$

Step-by-Step Solution

Answer

The roots of the equation \(2 x^{3}-11 x^{2}-x+30=0\) are 2, 3, and -5.

Step 1: List the factors of the constant and leading coefficient

The factors of the constant term (30) are 1, 2, 3, 5, 6, 10, 15, and 30, and their negatives. The factors of the leading coefficient (2) are 1 and 2. These will be used to create the possible rational roots.

Step 2: Create a list of possible rational roots

To get the possible rational roots \( p/q \), make a fraction where \( p \) is a factor of 30 and \( q \) is a factor of 2. This gives the set \{±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30, ±1/2, ±3/2, ±5/2, ±15/2\}.

Step 3: Test each possible root

Plug each possible root into the equation and check if it makes the equation true. In this case, you'll find that the roots are 2, 3, and -5.

Key Concepts

Rational roots

Rational roots are solutions to a polynomial equation that can be expressed as a fraction or ratio of two integers. These solutions are called "rational" because they may have a denominator other than one. Identifying rational roots can be a helpful way of solving polynomial equations, especially when dealing with equations involving higher degree polynomials. The Rational Root Theorem is a useful tool for finding these solutions.

The Rational Root Theorem states that if a polynomial has a rational root expressed as a fraction \( \frac{p}{q} \), then \( p \) is a factor of the polynomial's constant term, and \( q \) is a factor of the polynomial's leading coefficient. This theorem allows us to systematically list potential rational roots and test them one by one to find actual roots that satisfy the equation.

Polynomial equations

Polynomial equations involve expressions that include variables, coefficients, and exponents. These equations take the general form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0 \) where each \( a_i \) represents a coefficient and the highest exponent \( n \) is the degree of the polynomial. Solving polynomial equations means finding values of \( x \) that make the equation true.

Polynomial equations can have different types of solutions, including:

• Rational roots: as previously discussed, are expressed as fractions.
• Irrational roots: involve square roots and cannot be simplified into a fraction.
• Imaginary roots: occur in polynomials where no real solution exists, involving the imaginary unit \( i \).

Using methods like factoring, the Rational Root Theorem, or synthetic division helps find these solutions effectively.

Factors of coefficients

Factors of coefficients are integral to identifying potential rational roots using the Rational Root Theorem. In the context of a polynomial equation, the coefficients are the numerical values in front of each term's variables. These coefficients are crucial in determining the initial set of potential rational roots.

When applying the Rational Root Theorem, we first need to identify:

• Factors of the constant term: This is the last term in the polynomial (without any variables). Its factors are numbers that can divide it evenly without a remainder.
• Factors of the leading coefficient: This is the coefficient in front of the term with the highest power of the variable. Like the constant term, its factors are numbers that can divide it evenly.

By combining factors of the constant term over factors of the leading coefficient, we create a set of possible rational roots that can be tested to find real solutions to the polynomial equation.