Answer :
After evaluating the expression f(x) - 9f(x) - g(x), we obtain -9x⁷ + x⁶ - 7x³ + 5x² + 15, which represents the simplified result of the given functions. This expression delineates the polynomial combination of f(x) and g(x), offering insight into their relationship through subtraction.
Step 1: Substitute the functions f(x) and g(x) into the expression f(x) - 9f(x) - g(x).
- f(x) = -3x⁷ - x⁶ + 2x³ - 6
- g(x) = -3x⁷ - 5x³ - 5x² + 9
Step 2: Multiply f(x) by 9.
- 9f(x) = 9(-3x⁷ - x⁶ + 2x³ - 6)
- 9f(x) = -27x⁷ - 9x⁶ + 18x³ - 54
Step 3: Rewrite the expression as f(x) - 9f(x) - g(x).
- f(x) - 9f(x) - g(x) = (-3x⁷ - x⁶ + 2x³ - 6) - (-27x⁷ - 9x⁶ + 18x³ - 54) - (-3x⁷ - 5x³ - 5x² + 9)
Step 4: Simplify each term separately.
- f(x) - 9f(x) - g(x) = -3x⁷ - x⁶ + 2x³ - 6 + 27x⁷ + 9x⁶ - 18x³ + 54 - (-3x⁷ - 5x³ - 5x² + 9)
Step 5: Combine like terms.
- f(x) - 9f(x) - g(x) = -3x⁷ + 27x⁷ - x⁶ + 9x⁶ + 2x³ - 18x³ - 5x³ - 5x² - 6 + 54 - 9
Step 6: Simplify further.
- f(x) - 9f(x) - g(x) = -9x⁷ + x⁶ - 7x³ - 5x² + 15
Thus, the simplified expression for f(x) - 9f(x) - g(x) is -9x⁷ + x⁶ - 7x³ + 5x² + 15.
Complete Question:
Considering the functions f(x) = -3x^7 - x^6 + 2x^3 - 6 and g(x) = -3x^7 - 5x^3 - 5x^2 + 9, determine and simplify the expression f(x) - 9f(x) - g(x).
