Answer
1. f + g = 3x² - 7x + 14
2. f - g = x² - 7x - 4
3. f.g = (x - 1)(2x -5)(x² + 9)
4. f/g = 2 -7x -13/x² + 9
Step-by-step explanation
Given functions:
f(x) = 2x² - 7x + 5
g(x) = x² + 9
1. f + g = (2x² - 7x + 5) + (x² + 9)
= 2x² - 7x + 5 + x² + 9
= 2x² + x² - 7x + 5 + 9
= 3x² - 7x + 14
2. f - g = (2x² - 7x + 5) - (x² + 9)
= 2x² - 7x + 5 - x² - 9
= 2x² - x² - 7x + 5 - 9
= x² - 7x - 4
3. f.g
= (2x² - 7x + 5)(x² + 9)
= 2x²(x² + 9) - 7x(x² + 9) + 5(x² + 9)
= 2x⁴ + 18x²- 7x³ - 63x + 5x² + 45
= 2x⁴ - 7x³ + 18x² + 5x² - 63x + 45
= 2x⁴ - 7x³ + 23x² - 63x + 45
Using the rational root theorem
= (x - 1) 2x⁴ - 7x³ + 23x² - 63x + 45/x - 1
= 2x⁴ - 7x³ + 23x² - 63x + 45/x-1
= 2x³ - 5x² + 18x - 45
= (x -1)(2x³ - 5x² + 18x - 45)
Now, fatorising 2x³ - 5x² + 18x - 45 yields
= (2x -5)(x² + 9)
Hence, 2x⁴ - 7x³ + 23x² - 63x + 45 = (x - 1)(2x -5)(x² + 9)