other types of equations polynomial equations. factoring solve
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Other types of Equations
Polynomial Equations
Factoring
• Solve 24 483 xx
Factoring
• Solve
• First, set the equation equal to zero.
24 483 xx
0483 24 xx
Factoring
• Solve
• First, set the equation equal to zero.
• Take out the common factor.
24 483 xx
0483 24 xx
0)16(3 22 xx
Factoring
• Solve
• First, set the equation equal to zero.
• Take out the common factor. Factor.
24 483 xx
0483 24 xx
0)16(3 22 xx 0)4)(4(3 2 xxx
Factoring
• Solve
• First, set the equation equal to zero.
• Take out the common factor. Factor. Solve.
24 483 xx
0483 24 xx
0)16(3 22 xx 0)4)(4(3 2 xxx
0
03 2
x
x4
04
x
x
4
04
x
x
Factoring
• You try: xx 82 3
Factoring
• You try: xx 82 3
)2)(2(2
0)4(2
0822
3
xxx
xx
xx
Factoring
• You try: xx 82 3
)2)(2(2
0)4(2
0822
3
xxx
xx
xx
0
02
x
x
2
02
x
x
2
02
x
x
Factor by grouping
• When you have a 3rd degree equation with four terms, you will factor by grouping.
0933 23 xxx
Factor by grouping
• When you have a 3rd degree equation with four terms, you will factor by grouping.
0933 23 xxx
0)3(3)3(2 xxx
Factor by grouping
• When you have a 3rd degree equation with four terms, you will factor by grouping.
0933 23 xxx
0)3(3)3(2 xxx
0)3)(3( 2 xx
3
03
x
x
3
3
032
2
ix
x
x
Factor by grouping
• You try: 0632 23 xxx
Factor by grouping
• You try: 0632 23 xxx
0)3)(2(
0)2(3)2(2
2
xx
xxx
Factor by grouping
• You try: 0632 23 xxx
0)3)(2(
0)2(3)2(2
2
xx
xxx
2
02
x
x
3
3
032
2
ix
x
x
Solving a 4th degree
• We will look at a 4th degree equation just like we did a 2nd degree equation. Now, instead of starting your factoring with an , it will be an .2xx
Solving a 4th degree
• We will look at a 4th degree equation just like we did a 2nd degree equation. Now instead of starting your factoring with an , it will be an .
• Solve
2xx
023 24 xx
Solving a 4th degree
• We will look at a 4th degree equation just like we did a 2nd degree equation. Now instead of starting your factoring with an , it will be an .
• Solve
2xx
023 24 xx
0_)_)(( 22 xx
0)1)(2( 22 xx
Solving a 4th degree
• We will look at a 4th degree equation just like we did a 2nd degree equation. Now instead of starting your factoring with an , it will be an .
• Solve
2xx
023 24 xx
0_)_)(( 22 xx
0)1)(2( 22 xx
0)1)(1)(2( 2 xxx
Solving a 4th degree
• We will look at a 4th degree equation just like we did a 2nd degree equation. Now instead of starting your factoring with an , it will be an .
• Solve
2xx
023 24 xx
0_)_)(( 22 xx
0)1)(2( 22 xx
0)1)(1)(2( 2 xxx
2
2
022
2
x
x
x
1,01
1,01
xx
xx
Solving a 4th degree
• You try: 0365 24 xx
Solving a 4th degree
• You try: 0365 24 xx
0)4)(9( 22 xx
Solving a 4th degree
• You try: 0365 24 xx
0)4)(3)(3(
0)4)(9(2
22
xxx
xx
Solving a 4th degree
• You try: 0365 24 xx
0)4)(3)(3(
0)4)(9(2
22
xxx
xx
3
03
x
x
3
03
x
x
ix
x
x
2
4
042
2
Equations involving radicals
• Solve: 272 xx
Equations involving radicals
• Solve:
• First, get the radical by itself.
272 xx
272 xx
Equations involving radicals
• Solve:
• First, get the radical by itself.
• Next, square both sides.
272 xx
272 xx
4472 2 xxx
Equations involving radicals
• Solve:
• First, get the radical by itself.
• Next, square both sides.
• Solve.
272 xx
272 xx
4472 2 xxx
0322 xx
Equations involving radicals
• Solve:
• First, get the radical by itself.
• Next, square both sides.
• Solve.
272 xx
272 xx
4472 2 xxx
0322 xx )1)(3(0 xx 1
3
x
x
Equations involving radicals
• Solve:
• Now, you have to be careful. By squaring both sides, you may introduce an extraneous solution. You must check each solution.
272 xx
Equations involving radicals
• Solve:
• Now, you have to be careful. By squaring both sides, you may introduce an extraneous solution. You must check each solution.
272 xx
1
3
x
x
231
2)3(7)3(2
213
2)1(7)1(2
Equations involving radicals
• Solve:
• The only solution to this equation is .
272 xx
1x
Equations involving radicals
• You try: 33 xx
Equations involving radicals
• You try: 33 xx
1,6
)1)(6(0
670
963
)3()3(
2
2
22
x
xx
xx
xxx
xx
Equations involving radicals
• You try:
• You need to check each answers.
33 xx
1,6
)1)(6(0
670
963
)3()3(
2
2
22
x
xx
xx
xxx
xx
3141
3636
Equations involving radicals
• You try:
• You need to check each answers. The only answer is x = 6.
33 xx
1,6
)1)(6(0
670
963
)3()3(
2
2
22
x
xx
xx
xxx
xx
3141
3636
Equations involving radicals
• You try: 035 x
Equations involving radicals
• You try: 035 x
35 x
4
95
x
x
Equations involving radicals
• You try:
• Check your answer.• The answer works.
035 x
35 x
4
95
x
x
03)4(5
Absolute Value
• Absolute value represents distance from zero, that is why it is always positive. There will be two answers since we can go to the right from zero or to the left. 3 and -3 are both the same distance from zero.
Absolute Value
• Solve: 3|| x
Absolute Value
• Solve:
• The answer is , since they are both 3 units from zero.
3|| x
3
Absolute Value
• Solve: 4|7| x
Absolute Value
• Solve:
• We want to be 4 units from zero. There are two solutions.
4|7| x
7x
Absolute Value
• Solve:
• We want to be 4 units from zero. There are two solutions.
4|7| x
7x
3
47
x
x
11
47
x
x
Absolute Value
• You try: 9|52| x
Absolute Value
• You try: 9|52| x
7
142
952
x
x
x
2
42
952
x
x
x
Absolute Value
• Solve: 64|3| 2 xxx
Absolute Value
• Solve:
• We need to solve two equations.
64|3| 2 xxx
6432 xxx )64(32 xxx
Absolute Value
• Solve:
• We need to solve two equations.
64|3| 2 xxx
2,3
0)2)(3(
06
6432
2
xx
xx
xx
xxx
1,6
0)1)(6(
067
6432
2
xx
xx
xx
xxx
Absolute Value
• Solve:
• We need to solve two equations.
• We need to now check our answers.
64|3| 2 xxx
2,3
0)2)(3(
06
6432
2
xx
xx
xx
xxx
1,6
0)1)(6(
067
6432
2
xx
xx
xx
xxx
Absolute Value
• Solve:
• We need to solve two equations.
• We need to now check our answers.• Only -3 and 1 will work.
64|3| 2 xxx
2,3
0)2)(3(
06
6432
2
xx
xx
xx
xxx
1,6
0)1)(6(
067
6432
2
xx
xx
xx
xxx
Homework
• Pages 140-141• 1-13 odd• 29,31,65,66
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