ACTIVITY 41 Review For Final Problem 17 Evaluate the expression:
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Transcript of ACTIVITY 41 Review For Final Problem 17 Evaluate the expression:
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ACTIVITY 41
Review For Final
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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:
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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
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11
1
11
122
2
xx
xx
xx
x
11
112
2
xx
xxx
11
12
22
xx
xxx
11
12
xx
x
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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
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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
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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
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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
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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
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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
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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
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952 x
952 and 952 xx5 5 5 5
142 x 42 x
2
14x 7
2
4x 2
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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,
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112 x
112 x Or 112 x02 x
2
0x
0x
22 x
2
2x
1x
1 0 ,01,
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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
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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
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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
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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
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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
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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
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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
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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