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True [87]
3 years ago
6

These roots of the polynomial equation x^4-4x^3-2x^2+12x+9=0 are 3,-1,-1. Explain why the fourth root must be a real number. Fin

d the fourth root.
Mathematics
2 answers:
Svetradugi [14.3K]3 years ago
8 0

Answer:

Must be real because non real roots occur in pairs.

4th root = 3

Step-by-step explanation:

It must be real because complex roots occur only in conjugate pairs.

because of the  + 9  in the polynomial the 4th root will be 9 / (3*-1*-1) = 9/3 = 3

Alex787 [66]3 years ago
4 0

Roots with imaginary parts always occur in conjugate pairs. Three of the four roots are known and they are all real, which means the fourth root must also be real.

Because we know 3 and -1 (multiplicity 2) are both roots, the last root r is such that we can write

x^4-4x^3-2x^2+12x+9=(x-3)(x+1)^2(x-r)

There are a few ways we can go about finding r, but the easiest way would be to consider only the constant term in the expansion of the right hand side. We don't have to actually compute the expansion, because we know by properties of multiplication that the constant term will be (-3)(1)(1)(-r)=3r.

Meanwhile, on the left hand side, we see the constant term is supposed to be 9, which means we have

3r=9\implies r=3

so the missing root is 3.

Other things we could have tried that spring to mind:

- three rounds of division, dividing the quartic polynomial by (x-3), then by (x+1) twice, and noting that the remainder upon each division should be 0

- rational root theorem

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3 years ago
Which is the graph of the given function?
Schach [20]

ANSWER

The correct choice is A.

EXPLANATION

The given cosine function is:

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This function has an amplitude of 2.

Hence it must have range,

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It has a period of

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Hence the function must complete one cycle on the interval,

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The negative sign shows a reflection in the y-axis but since the cosine function is even, the function retraces itself.

The correct choice is A

4 0
3 years ago
5 roommates share the cost of an apartment: rent,electricity, and gas. rent costs 1800 and gas costs 150 per month. each roommat
Naddik [55]

Answer:

Electricity bill is 175

<em />

Step-by-step explanation:

Given

Rent = 1800

Gas = 150

Each\ Roommate = 425 monthly

Number\ of\ Roommate = 5

Required

Determine the electricity bill

If each roommate pays 425, then 5 roommates pay 5 * 425

Total = 5 * 425

Total = 2125

This implies that: the monthly cost of room, gas and electricity add up to 2125.

In other words,

Gas + Rent + Electricity = 2125

150 + 1800+ Electricity = 2125

1950+ Electricity = 2125

Electricity = 2125 - 1950

Electricity = 175

<em>Hence, the electricity bill is 175</em>

4 0
2 years ago
Which represents the volume of the cylinder, in cubic
11Alexandr11 [23.1K]

Answer: 300π

Step-by-step explanation: Volume=(pi)(radius^2)(height)

Volume=(pi)(5^2)(12)

V=(pi)(25)(12)

V=(pi)(300)

6 0
3 years ago
Find the mean, median, and mode of the data. Which reflects the best measure of the center? 43, 38, 37, 57, 57, 58, 45
Allushta [10]

ANSWER:

mean:47.8

mode:57

median:45

37, 38, 43, 45, 57, 57, 58

8 0
3 years ago
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