MAT 112 -03/04 Test #3 Key

1. Find a polynomial with integer coefficients that has zeros: 2, -1, 2 + i, 2 - i.

(x - 2) (x + 1) (x - (2 + i)) (x - (2 - i)

(x - 2) (x + 1) (x2 - 4x + 5)

(x - 2) (x3 - 3x2 + x + 5)

x4 - 3x3 + x2 + 5x - 2x3 + 6x2 - 2x - 10

x4 - 5x3 + 7x2 + 3x - 10

2. Write the polynomial (a) as the product of factors that are irreducible over the reals and (b) in completely factored form: (Hint: x - sqrt(3), is a factor.)
f(x) = 4x4 - 3x2 - 27.

The hint and graphing reveals + sqrt(3) are good candidates for real zeros. 

sqrt(3)|4       0  -3       0  -27
	  4sqrt(3) 12 9sqrt(3)  27
	4 4sqrt(3)  9 9sqrt(3)   0
 
-sqrt(3)|  4  4sqrt(3)   9   9sqrt(3)
	     -4sqrt(3)   0  -9sqrt(3)
	  4         0    9         0

This last qoutient can be written as 4x2 + 9 = 0.
Solving gives us the complex factors: + 3/2 i.

(a.) (x + sqrt(3)) (x - sqrt(3)) ( 4x2 + 9) (b.) (x + sqrt(3)) (x - sqrt(3)) ( x + 3/2 i) ( x - 3/2 i)

3. The pitcher's mound is 60.5 feet from home plate and the distance between the bases is 90 feet. Solve the triangle

 

Let A = 45o, b = 60.5 ft., c = 90 ft.

a2 = b2 + c2 - 2bc cos A Þ a2 = 60.52 + 902 - 2 (60.5) (90) cos 45o = 4059.857

a = 63.717

cos B = (63.7172 + 902 - 60.5) / (2 (63.717) (90) ) = .741090036062

B = 42.82o

C = 180 - A - B = 92.82o.

 

 

4. Solve the triangle. If two solutions exist, find both. C = 32o 20', a = 12.8, c = 7.

Let h be the altitude, h = 12.8 sin(32 1/3o) = 6.846. Since h<c<a, we have the ambiguous case.

Case 1: A = arcsin(12.8 (sin(32 1/3o) / 7) = 77.96o, B = 180 - A - C = 69.707o,
b = sin B (c / sin C) = 12.28.

Case 2: A2 = 180 - A = 102.04o, B2 = 180 - A2 - C = 45.63o,
b2 = sin B2 (c / sin C) = 9.36.

 5. Use DeMoivre's Theorem to find the indicated power of the complex number. express the result in standard form.

3(3 - 2i)6

We have 2z6, with z = 3 - 2i. So r = |z| = sqrt(32 + (-2)2) = sqrt(13),
q = arctan(b/a) = arctan(-2/3) = -33.69007o.

3z7 =

3[136/2 (cos(6(-33.69007o) + i sin(6(-33.69007o) )]

=

3[133 ( cos(-202.14o) + i sin(-202.14o) )

=

3[2197 ( -0.926263 + i 0.37687756)

=

-6105 + 2484 i

 6. Write an expression for the most apparent nth term of the sequence. (Assume n begins with 1.)

1/3, 8/4, 27/5, 64/ 6, 125/7, ...

Is this sequence arithmetic, geometric, or neither?

Numerator

1

8

27

64

125

...

n3

 

Denominator

3

4

5

6

7

...

n + 2

Index, n:

1

2

3

4

5

 

 

 

an = n3 / (n + 2)

a2 - a1 = 5/3 ¹ a3 - a2 = 17/5, not arithmetic, a2/a1 = 6 ¹ a3/a2 = 27/10, not geometric, so neither.

 7. A brick patio has the approximate shape of a trapezoid. The patio has 21 rows of bricks. The first row has 10 bricks, the second row 16 bricks, and the 21st row has 130 bricks. How many bricks are used to make the patio.

a1

a2

...

a21

10

16

...

130

An arithmetic sequence with common difference, d = 6. c = a1 - d = 10 - 6 = 4.
an = dn + c Þ 130 = 6n + 4, n = 126/6 = 21.

S21 = 21/2 (10 + 130) = 21(70) = 1470.

8. Use summation notation to express the sum, then calculate the value of the sum.

-12 - 9 - 6 - 3 - 0 + 3 + 6 + ... + 300

An arithmetic sequence with d = 3,
an = a1 + (n - 1)d Þ 300 = -12 + (n - 1)3
315 = 3n
n = 105

 

 

We can test this result:
a105 = -12 + (105 - 1)3 = 300

9. Represent the complex number graphically, and find the standard form of the number.

5(cos 210o + isin210o)

 

a = 5 cos 210o = -5 sqrt(3)/2, b = 5 sin 210o = -5/2,

z = -5 sqrt(3)/2 - 5/2 i.