Three roots of the polynomial equation x - 4x3 - 2x2 + 12x + 9 = 0 are 3, -1 and -1. explain why the fourth root must be a
real number. find the fourth root.

The root must be -3 and be a real number.

Step-by-step explanation:

Since the roots are 3, -1, -1 and x that means the equation can be converted into a(bx-3)(cx+1)(dx+e). -3 * 1 * e = 9. This means that e = -3. Since -3 belongs to the reals, that must mean the last root must be a real number.

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step-by-step explanation:

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt