The polynomial of the fourth degree that satisfies the zeros given as well as the condition for the coefficient of x³ is;
f(x) = -2x⁴ + 4x³ + 58x² - 60x
- We are given the four zeros of the polynomial as; -5, 0, 1, 6. These four numbers mean they are the x-intercepts of the polynomial and as such the factors of the polynomial will be;
(x + 5), x, (x - 1), (x - 6)
- The polynomial will be gotten by multiplying these factors by themselves. Thus;
f(x) = (x + 5) * x * (x - 1) * (x - 6)
f(x) = (x² + 5x) × (x² - 7x + 6)
f(x) = x⁴ - 7x³ + 6x² + 5x³ - 35x² + 30x
f(x) = x⁴ - 2x³ - 29x² + 30x
- We are told that the coefficient of x³ is 4 but what we have is -2. Thus, we will multiply each term by -2 to make the coefficient of x³ to be 4. Thus, we now have;
f(x) = -2x⁴ + 4x³ + 58x² - 60x
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