Find a polynomial f(x) of degree 5 that has the following zero -4, 8, -3, 1,9

f(x) = x^5 - 11x^4 - 25x³ + 335x² + 564x - 864

Step-by-step explanation:

The factors are (x + 4) , (x - 8) , (x + 3) , ( x - 1) , (x - 9)

(x + 4) ( x - 8) = x² + 4x - 8x - 32 = x² - 4x - 32

(x + 3) (x² - 4x - 32) = x³ - 4x² - 32x + 3x² -12x - 96 = x³ - x² - 44x -96

(x - 1) (x³ - x² - 44x -96) = x^4 - x³ - 44x² - 96x - x³ + x² + 44x + 96

= x^4 - 2x³ - 43x² - 52x + 96

(x - 9) (x^4 - 2x³ - 43x² - 52x + 96) = x^5 - 2x^4 - 43x³ - 52x² + 96x - 9x^4 + 18x³ + 387x² + 468x - 864

= x^5 - 11x^4 - 25x³ + 335x² + 564x - 864

∴ f(x) = x^5 - 11x^4 - 25x³ + 335x² + 564x - 864

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