Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.
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Transcript of Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.
![Page 1: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/1.jpg)
example 4 Solving a Quartic Equation
Chapter 6.4
Solve the equation .4 3 22 10 13 6 0x x x x
2009 PBLPathways
![Page 2: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/2.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
![Page 3: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/3.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
1.Determine the possible rational solutions of f(x) = 0.
2.Graph y = f(x) to see if any of the values from Step 1 are x-intercepts. The x-intercepts are also solutions to f(x) = 0.
3.Find the factors associated with the x-intercepts from Step 2.
4.Use synthetic division to divide f (x) by the factors from Step 3 to confirm the graphical solutions and find additional factors. Continue until a quadratic factor remains.
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
![Page 4: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/4.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
1.Determine the possible rational solutions of f(x) = 0.
1, 2, 3, 6
and
1 2 3 6 , , ,
2 2 2 2
Factors of 6 1, 2, 3, 6
Factors of 2 1, 2
![Page 5: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/5.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
1.Determine the possible rational solutions of f(x) = 0.
1, 2, 3, 6
and
1 2 3 6 , , ,
2 2 2 2
Factors of 6 1, 2, 3, 6
Factors of 2 1, 2
![Page 6: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/6.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
1.Determine the possible rational solutions of f(x) = 0.
1, 2, 3, 6
and
1 2 3 6 , , ,
2 2 2 2
Factors of 6 1, 2, 3, 6
Factors of 2 1, 2
![Page 7: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/7.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
2.Graph y = f(x) to see if any of the values from Step 1 are x-intercepts. The x-intercepts are also solutions to f(x) = 0.
x
y
1, 2, 3, 6
and
1 2 3 6 , , ,
2 2 2 2
Factors of 6 1, 2, 3, 6
Factors of 2 1, 2
![Page 8: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/8.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
2.Graph y = f(x) to see if any of the values from Step 1 are x-intercepts. The x-intercepts are also solutions to f(x) = 0.
x
y
1, 2, 3, 6
and
1 2 3 6 , , ,
2 2 2 2
Factors of 6 1, 2, 3, 6
Factors of 2 1, 2
(-2, 0)
(-1, 0)
![Page 9: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/9.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
3.Find the factors associated with the x-intercepts from Step 2.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x ?
![Page 10: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/10.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
4.Use synthetic division to divide f (x) by the factors from Step 3 to confirm the graphical solutions and find additional factors. Continue until a quadratic factor remains.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x ?
1 2 10 13 1 6
2 8 5 6
2 8 5 6 0
2 2 8 5 6
4 8 6
2 4 3 0
3 22 8 5 6x x x ?
![Page 11: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/11.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
4.Use synthetic division to divide f (x) by the factors from Step 3 to confirm the graphical solutions and find additional factors. Continue until a quadratic factor remains.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x ?
1 2 10 13 1 6
2 8 5 6
2 8 5 6 0
2 2 8 5 6
4 8 6
2 4 3 0
3 22 8 5 6x x x
![Page 12: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/12.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
4.Use synthetic division to divide f (x) by the factors from Step 3 to confirm the graphical solutions and find additional factors. Continue until a quadratic factor remains.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x ?
1 2 10 13 1 6
2 8 5 6
2 8 5 6 0
2 2 8 5 6
4 8 6
2 4 3 0
![Page 13: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/13.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
4.Use synthetic division to divide f (x) by the factors from Step 3 to confirm the graphical solutions and find additional factors. Continue until a quadratic factor remains.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
1 2 10 13 1 6
2 8 5 6
2 8 5 6 0
2 2 8 5 6
4 8 6
2 4 3 0
![Page 14: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/14.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
22 4 3 0x x
24 4 4 2 3
2 2
4 40
4
2 10
2
x
![Page 15: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/15.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
22 4 3 0x x
24 4 4 2 3
2 2
4 40
4
2 10
2
x
2 4
2
b b acx
a
![Page 16: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/16.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
22 4 3 0x x
24 4 4 2 3
2 2
4 40
4
2 10
2
x
![Page 17: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/17.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
22 4 3 0x x
24 4 4 2 3
2 2
4 40
4
2 10
2
x
![Page 18: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/18.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
4 3 2 22 10 13 6 ( 1)( 2)( 2 4 3 )x x x x x x x x
22 4 3 0x x
24 4 4 2 3
2 2
4 40
4
2 10
2
x
![Page 19: Example 4 Solving a Quartic Equation Chapter 6.4 Solve the equation. 2009 PBLPathways.](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d095503460f949dae5a/html5/thumbnails/19.jpg)
2009 PBLPathways
Solve the equation .4 3 22 10 13 6 0x x x x
Solving Cubic and Quartic Equations of the Form f(x) = 0
5.Use factoring or the quadratic formula to find the solutions associated with the quadratic factor. These solutions are also solutions to f(x) = 0.
20 ( 1)( 2)( 2 4 3 )x x x x
2 10 2 102, 1, ,
2 2x
x
y
(-2, 0)
(-1, 0)
(0.58, 0)(-2.58, 0)