example 4 Solving a Quartic Equation

Chapter 6.4

Solve the equation .4 3 22 10 13 6 0x x x x

2009 PBLPathways

2009 PBLPathways

Solve the equation .4 3 22 10 13 6 0x x x x

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.

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

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

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

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

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)

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 ?

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 ?

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

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

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

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

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

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

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

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

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)