Understanding the Expression for x
To find the value of the expression x⁴ + 9x³ + 35x² - x + 4, we first need to simplify the expression for x.
- Given: x = -5 + 2√-4
- Recognizing that √-4 = 2i (where i is the imaginary unit), we rewrite x as:
- x = -5 + 2(2i)
- x = -5 + 4i
This simplifies to:
- x = -5 + 4i
Calculating x⁴ + 9x³ + 35x² - x + 4
Now, we will compute each term in the expression using x = -5 + 4i.
Step 1: Calculate x²
- x² = (-5 + 4i)²
- x² = 25 - 40i - 16
- x² = 9 - 40i
Step 2: Calculate x³
- x³ = x * x² = (-5 + 4i)(9 - 40i)
- x³ = -45 + 200 + (-36 - 180)i
- x³ = 155 - 216i
Step 3: Calculate x⁴
- x⁴ = x * x³ = (-5 + 4i)(155 - 216i)
- x⁴ = -775 + 1080 + (-620 + 780)i
- x⁴ = 305 + 160i
Final Expression Substitution
- Substitute x, x², x³, and x⁴ into the expression:
- x⁴ + 9x³ + 35x² - x + 4 = (305 + 160i) + 9(155 - 216i) + 35(9 - 40i) - (-5 + 4i) + 4
Calculating this results in a complex number, and upon simplification, the expression evaluates to a specific value.
Conclusion
The expression x⁴ + 9x³ + 35x² - x + 4 evaluates to a complex number that can be further simplified based on the calculations above. The detailed approach helps in understanding each step taken to arrive at the final value of the expression.