Given [tex]f(x) = -3x^7 - x^6 + 2x^3 - 6[/tex] and [tex]g(x) = -3x^7 - 5x^3 - 5x^2 + 9[/tex], find and simplify [tex]f(x) - 9f(x) - g(x)[/tex].

Question

13-05-2024
Mathematics

High School

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Given [tex]f(x) = -3x^7 - x^6 + 2x^3 - 6[/tex] and [tex]g(x) = -3x^7 - 5x^3 - 5x^2 + 9[/tex], find and simplify [tex]f(x) - 9f(x) - g(x)[/tex].

Answer :

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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).