Solve by Factoring:. Solve by completing the Square:
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Transcript of Solve by Factoring:. Solve by completing the Square:
Solve by completing the Square:
2( ) 3 6 1f x x x
3
321
3
321
3
41
3
41
134
12331
231
631
2
2
2
2
2
2
x
x
x
x
x
xx
xx
xx
Remainder Theorem
Remainder = f(k) Example:
f(2)=
3 23 2 5 5 2x x x x
31
510824
5104283
5252223 23
2686
55232
311343
Find the remainder:
3 28 2 5 4 2x x x x
86
410864
4104288
42522282 23
f
Therefore, x = -2 is NOT a root! FactorTheorem
Rational Zero Test
If f(x)=anxn + an-1xn-1 +… + a1x + a0
Then the possible rational roots are
Factors of the last term (a0) over the factors of the first term (an)
termfirstoffactors
termlastoffactors
Find all real roots:3 2( ) 1f x x x x
11
1:
Possible
1 1111
0101
101
11
111
11 2
xx
xxx
xx
Mult. of 2Touches.
Goes Through
x y
0 1
Find all real roots:
1 3832
0352
352
3121
3521 2
xxx
xxx
Goes Through ALL
x y
0 3
3 2( ) 2 3 8 3f x x x x
2
3,2
1,3,1
2,1
3,1:
Possible
3,2
11 xxx
Find all real roots:
1 65551
061161
61161
x y
0 -6
4 3 2( ) 5 5 5 6f x x x x x
6,3,2,11
6,3,2,1:
Possible
1 651
3211
6511
611612
23
xxxx
xxxx
xxxx
651
3211 xxxx
All Go Through
Find all real roots:Do NOT Graph.
1 63111
06321
6321
6,3,2,11
6,3,2,1:
Possible
2 602
321
63212
23
xxx
xxxx
301
4 3 2( ) 3 6f x x x x x
321 2 xxx
NOT Real!
Find all real roots:Do NOT Graph.
5 4 3 2( ) 3 5 15 4 12f x x x x x x
12,6,4,3,2,11
12,6,4,3,2,1:
Possible
1
01216141
1216141
12415531
1012431
12431
2
0651
12102
32211
65211
124311
121641
2
23
234
xxxxx
xxxxx
xxxxx
xxxxx
32211 xxxxx
Complex Numbers:
Consists of a real number plus an imaginary number
Looks like: a + bi
Can also be called an imaginary number
If a = 0, then it’s a pure imaginary number