Final answer:
Begin with listing possible rational zeros using the Rational Zero Theorem. Then, use synthetic division to test these zeros to find an actual zero. Finally, use the zero to factor and solve the remaining quadratic to find the full solution set.
Step-by-step explanation:
To answer the equation 8x³ + 34x² - 29x + 5 = 0, we will first list all possible rational zeros using the Rational Zero Theorem. The possible rational zeros are the p/q values, where p is a factor of the constant term (5), and q is a factor of the leading coefficient (8).
All Rational Zeros: Possible values for p are ±1, ±5, and for q are ±1, ±2, ±4, ±8. Combining these, we can have possible rational zeros of ±1, ±1/2, ±1/4, ±1/8, -1, -1/2, -1/4, -1/8, ±5, ±5/2, ±5/4, ±5/8, -5, -5/2, -5/4, -5/8.
Next, we use synthetic division to test these possible zeros. When we find one that gives a remainder of 0, we have found an actual zero of the equation. For example, if we find that x = 1 is a zero, we can then divide the polynomial by (x - 1) to find a quadratic equation. This quadratic can then be solved using the quadratic formula.
Once we have identified one rational zero and solved the quadratic, we will have the full set of solutions or roots for the cubic equation.