Question

Answer the following questions about the equation below. 8x³ - 46x² + 31x - 5 = 0

a) List all rational roots that are possible according to the Rational Zero Theorem.
a. ±1, ±2, ±4, ±8, ±1/5, ±2/5, ±4/5, ±8/5
b. ±1, ±2, ±4, ±8
c. ±1, ±5, ±1/2, ±5/2, ±1/4, ±5/4, ±1/8, ±5/8
d. ±1, ±5

b) Use synthetic division to test several possible rational roots in order to identify one actual root.

c) Use the root from part (b) to solve the equation.

Answer 1

The possible rational roots for the equation 8x³ - 46x² + 31x - 5 = 0 are ±1, ±2, ±4, ±8, ±1/5, ±2/5, ±4/5, ±8/5. To find an actual root, synthetic division can be used to test possible roots. The root obtained can then be used to solve the equation through factoring.

Step-by-step explanation:

The Rational Zero Theorem states that if a polynomial has a rational root, it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. For the given equation 8x³ - 46x² + 31x - 5 = 0, the possible rational roots are ±1, ±2, ±4, ±8, ±1/5, ±2/5, ±4/5, ±8/5 (option a).

To test these possible roots, we can use synthetic division. Starting with one of the rational roots, such as 1, we divide the polynomial by (x - 1) using synthetic division. If the remainder is 0, then 1 is an actual root. In this case, we find that (x - 1) is a factor of the polynomial.

Using the root obtained from part (b), we can solve the equation by factoring. Since (x - 1) is a factor, we can rewrite the equation as (x - 1)(8x² - 38x + 5) = 0. Setting each factor equal to zero gives us the two possible solutions: x = 1 and 8x² - 38x + 5 = 0.