Sect. 5.1 Finding Zeros using Rational Root Theorem

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that if a polynomial has rational roots, they can be expressed as the ratio of factors of the constant term to factors of the leading coefficient.

Front

How do you find possible rational roots using the Rational Root Theorem?

Back

List all factors of the constant term and the leading coefficient, then form all possible fractions using these factors.

Front

Given the polynomial 2x^3 - 11x^2 + 12x + 9, what are the possible rational roots?

Back

±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

Front

What are the steps to apply the Rational Root Theorem?

Back

1. Identify the constant term and leading coefficient. 2. List their factors. 3. Form possible rational roots. 4. Test these roots in the polynomial.

Front

What is a zero of a polynomial?

Back

A zero of a polynomial is a value of x that makes the polynomial equal to zero.

Front

How can you verify if a number is a root of a polynomial?

Back

Substitute the number into the polynomial equation. If the result is zero, it is a root.

Front

What is the significance of finding zeros in polynomial equations?

Back

Finding zeros helps in graphing the polynomial and understanding its behavior, including x-intercepts.

Front

Solve for x: 2x^3 - 11x^2 + 12x + 9 = 0 using the Rational Root Theorem.

Back

The solutions are x = -1/2, x = 3 (with multiplicity 2).

Front

What is the complete list of possible rational zeros for f(x) = x^3 + 2x^2 - 6x + 8?

Back

±1, ±2, ±4, ±8.

Front

What are the x-intercepts of the polynomial x^3 - 3x^2 - 2x + 4?

Back

x = 1, x = 1 ± √5.