ACTIVITY 41

Review For Final

Problem 17

216

4 54

2 2 27 2

3 9

3 3

3

1

216

3 216

1

3 3332

1

3*2

1

6

1

Evaluate the expression:

Problem 83

Perform the indicated operations:

11

12

x

x

xSince there is no equal sign we find the LCD and the get each fraction over the LCD

11 LCD 2 xx

1

1

11

1

1

122

2

x

x

x

x

xx

x

11

1

11

122

2

xx

xx

xx

x

11

1

11

122

2

xx

xx

xx

x

11

112

2

xx

xxx

11

12

22

xx

xxx

11

12

xx

x

Problems 45, 49, 55, 59 and 65:

Factor the expressions:

22 62 xyyx xy2 yx 3

6384 23 xxx 24x 2x 3 2x

234 2 xxCan we factor 4x2+3 over the real numbers any further?

This is equivalent to asking if there are any real roots, and the descrimate tell us this!

acbD 4 Recall 2 34402 0

Consequently, there are no real roots; therefore, no factor over the reals

123 2 xx 133 2 xxxx3 1x 11 x

113 xx

1121 2 xx

1Let xw

122 ww

11 ww

21 w

211 x 2x

16 x

q

p 1

1

1possible rational zeros:

1 of divisors

1of divisors

11 0

1

1

1

1

1

1

1

1

1

0 0 0 0 1

1

1

1

0

)(xQ

112345 xxxxxx

16 x 112345 xxxxxx

11 1

1

1

0

0

1

10

0

1

1 1 1 1

10

)(xQ

16 x 112345 xxxxxx

1124 xxx 1x

11124 xxxx

16 x 11124 xxxx

So we need only factor x4+x2+1This does reduce further; however, we don’t have the tools to reduce this. So we have to look at the problem in a different way.

16 x 223 1 x

bababa 22

11 33 xx

2233 babababa 2233 babababa

111 32 xxxx

1111 22 xxxxxx

Problems 35, and 43

Find all real solutions of the following equations:

31

21

xx

Since there is an equal sign we find the LCD and multiply both sides by the LCD

1 x LCD x

1

211

xxxxLHS

1

21

11

xxx

xxx

xx 21

13 x 13 xxRHS xx 33 2

xxx 3313 2 x3 x3

xx 631 2 1 1

1630 2 xx

3a6b

1ca

acbbx

2

42

32

13466 2

6

12366

6

246

952 x

952 and 952 xx5 5 5 5

142 x 42 x

2

14x 7

2

4x 2

Problem 75, and 83.

Solve the following inequalities. Express the solution using interval notation an graph the solution set on the real number line.

01242 xx 026 xx

6 2

,26,

112 x

112 x Or 112 x02 x

2

0x

0x

22 x

2

2x

1x

1 0 ,01,

Problem 3

Consider the points P(–6, 2) and Q(4, –14)(a) Plot P and Q on the Coordinate plane.(b) Find the distance from P to Q.

2

12

2

12 yyxxPQ 22

)2()14()6(4 22

1664 22

1610

256100 356

Consider the points P(–6, 2) and Q(4, –14)(c) Find the midpoint of the segment PQ.

2,

22121 yyxx

M

2

142,

2

46M

2

12,

2

2M

6,1 M

Consider the points P(–6, 2) and Q(4, –14)(d) Sketch the line determined by P and Q, and find its equation in slope-intercept form

11 xxmyy

12

12

xx

yym

64

214

64

16

10

16

5

8

65

82

xy

65

82

xy

65

82

xy

5

48

5

82

xy

25

48

5

8

xy

5

5*

1

2

5

48

5

8

xy

5

10

5

48

5

8

xy

5

1048

5

8

xy

5

38

5

8

xy

Consider the points P(–6, 2) and Q(4, –14)(e) Sketch the circle that passes through Q and has center P, and fine the equation of this circle.

222 rkyhx

),( khC )2,6(r 356

222 35626 yx

35626 22 yx

Problem 40

Find an equation for the line that passes through the point (1,7) and is perpendicular to the line x – 3y +16 = 0.

yx 316 11 xxmyy

yx

3

16

3

16

3

1 xy

3

1m 3 m

137 xy

Problem 11

Show that the following equation represents a circle. Find its center and radius:

x2 + y2 + 2x – 6y + 9 = 0

yyxx 09 6 2 22

yyxx 019 6 12 22

yyxx 09199612 22

yx 091931 22

yx 0131 22

131 22 yx

222 rkyhx 1r)3,1(Center

Problem 11:

Solve the following system of equations:

xy

xxy

6

22

xxx 622

062 xx

023 xx

03 x 02 x3x 2x

36 y3y

3,3

26 y8y

8,2