Solving polynomial equation

Question

Assignment:

Q1. Factor completely, or state that the polynomial is prime.

2x³ + 250

Q2. Factor completely, or state that the polynomial is prime.

16x³ - 16x

Q3. Factor completely, or state that the polynomial is prime.

9x⁴ - 9

Q4. Factor by grouping. Assume any variable exponents represent whole numbers.

x³ + 9x - 4x² - 36

Q5. Factor by grouping. Assume any variable exponents represent whole numbers.

3x³ - 6x² + 7x - 14

Q6. First, write the value(s) that make the denominator(s) zero. Then solve the equation. (x - 8 / 2x + 5) + 5 = x + 6 / 5

Q7. Solve the equation. 2x / 5 = (x + 3) + 3

Q8.  Find the product and write the result in standard form.

(9 + 4i)(9 - 8i)

Q9.  Solve the polynomial equation by factoring and then using the zero product principle.

x³ + 9x² + 20x = 0

Q10.  Solve the radical equation, and check all proposed solutions.

√14x - 7=  x  + 3

Q11.  Solve the absolute value equation or indicate that the equation has no solution.

|x + 6| = 2

Q12.  Solve the equation by the square root property.
2x² = 26

Q13.  Solve the equation by the square root property.

(x - 6)² = 16

Q14. Solve the equation by expressing each side as a power of the same base and then equating exponents.

4^{(3x + 5)} = 1 / 256

Q15. Solve the equation by expressing each side as a power of the same base and then equating exponents.

9^{(x - 9) / 8} = √9

Q16. Solve the exponential equation. Express the solution set in terms of natural logarithms.

4^{x + 4} = 5 ^{2x + 5}

Q17. Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.
log₂(x + 2) + log₂(x - 4) = 4

Q18.  Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

log₆(x - 3) = 2

Q19.  Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

log₃(x + 5) = 2 + log₃(x - 1)

Q20. Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

log₉(6x + 2) = log₉(6x + 7)

Q21. Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log_b (xy⁶ / z³)

Q22. Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log₅(625 /√ x - 1)

Q23. Use properties of logarithms to write as a single logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

4log₃2 = 1 / 7 log₃(r - 3) - 1 / 2 log₃ r

Q24.  Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.

2x³ - 13x² + 22x - 8 = 0

Q25. Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.

3x⁴ - 19x³ + 69x² - 99x + 26 = 0

Provide complete and step by step solution for the question and show calculations and use formulas.