6 4 Factoring and Solving Polynomial Equations Factor
![6. 4 Factoring and Solving Polynomial Equations 6. 4 Factoring and Solving Polynomial Equations](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-1.jpg)
![Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Groupin g– Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Groupin g–](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-2.jpg)
![Solving Polynomial Equations The expressions on the previous slide are now equations: y = Solving Polynomial Equations The expressions on the previous slide are now equations: y =](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-3.jpg)
![Solve y = x 3 – 2 x 2 0 = x 2(x – Solve y = x 3 – 2 x 2 0 = x 2(x –](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-4.jpg)
![Solve y = x 4 – x 3 – 3 x 2 + 3 Solve y = x 4 – x 3 – 3 x 2 + 3](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-5.jpg)
![The Quadratic Formula For equations in quadratic form: ax 2 + bx + c The Quadratic Formula For equations in quadratic form: ax 2 + bx + c](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-6.jpg)
![Using the Quadratic Formula Solve the following cubic equation: Can this equation be to Using the Quadratic Formula Solve the following cubic equation: Can this equation be to](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-7.jpg)
![Factoring Sum or Difference of Cubes If you have a sum or difference of Factoring Sum or Difference of Cubes If you have a sum or difference of](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-8.jpg)
![Example Factor x 3 + 343. Note: This is a binomial. Are the first Example Factor x 3 + 343. Note: This is a binomial. Are the first](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-9.jpg)
![Example Factor 64 a 4 – 27 a = a(64 a 3 – 27) Example Factor 64 a 4 – 27 a = a(64 a 3 – 27)](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-10.jpg)
![Factor by Grouping Some four term polynomials can be factor by grouping. Example. Factor Factor by Grouping Some four term polynomials can be factor by grouping. Example. Factor](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-11.jpg)
![Example Factor 3 x 3 + 7 x 2 -12 x - 28 Step Example Factor 3 x 3 + 7 x 2 -12 x - 28 Step](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-12.jpg)
![Solving Polynomial Equations Solve Set equation equal to zero. Factor. Set each factor equal Solving Polynomial Equations Solve Set equation equal to zero. Factor. Set each factor equal](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-13.jpg)
- Slides: 13
![6 4 Factoring and Solving Polynomial Equations 6. 4 Factoring and Solving Polynomial Equations](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-1.jpg)
6. 4 Factoring and Solving Polynomial Equations
![Factor Polynomial Expressions In the previous lesson you factored various polynomial expressions Groupin g Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Groupin g–](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-2.jpg)
Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Groupin g– Common Factor factor th common terms a e first two nd then t two term he last s. Such as: 2(x – 2) 3 2 x x – 2 x = 3 – x 2 – 3 x + 3) 4 3 2 x(x x – 3 x + 3 x = 2(x – 1) – 3(x – 1)] x[x = Comm 2 – 3)(x – 1) on x(x = Factor
![Solving Polynomial Equations The expressions on the previous slide are now equations y Solving Polynomial Equations The expressions on the previous slide are now equations: y =](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-3.jpg)
Solving Polynomial Equations The expressions on the previous slide are now equations: y = x 3 – 2 x 2 and y = x 4 – x 3 – 3 x 2 +3 x To solve these equations, we will be solving for x when y = 0.
![Solve y x 3 2 x 2 0 x 2x Solve y = x 3 – 2 x 2 0 = x 2(x –](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-4.jpg)
Solve y = x 3 – 2 x 2 0 = x 2(x – 2) Let y = 0 Common factor Separate the factors and set them equal to zero. x 2 = 0 or x – 2 = 0 x=2 Solve for x Therefore, the roots are 0 and 2.
![Solve y x 4 x 3 3 x 2 3 Solve y = x 4 – x 3 – 3 x 2 + 3](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-5.jpg)
Solve y = x 4 – x 3 – 3 x 2 + 3 x 0 = x(x 3 – x 2 – 3 x + 3) 0 =x[x 2(x – 1) – 3(x – 1)] 0 = x(x – 1)(x 2 – 3) Let y = 0 Common factor Group Separate the factors and set them equal to zero. Solve for x x = 0 or x – 1 = 0 or x 2 – 3 = 0 x=1 x= Therefore, the roots are 0, 1 and ± 1. 73
![The Quadratic Formula For equations in quadratic form ax 2 bx c The Quadratic Formula For equations in quadratic form: ax 2 + bx + c](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-6.jpg)
The Quadratic Formula For equations in quadratic form: ax 2 + bx + c = 0, we can use the quadratic formula to solve for the roots of the equation. This equation is normally used when factoring is not an option.
![Using the Quadratic Formula Solve the following cubic equation Can this equation be to Using the Quadratic Formula Solve the following cubic equation: Can this equation be to](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-7.jpg)
Using the Quadratic Formula Solve the following cubic equation: Can this equation be to solve for x We still need y = x 3 + 5 x 2 – 9 x factored? here. Can this equation be 0 =YES x(xit 2 can + 5 x– – 9) factored? factor. x =common 0 x 2 + 5 x – 9 = 0 No. however, There are no twothe quadratic formula. We can, use integers that will multiply a = 1 to -9 and add to 5. b=5 c = -9 Remember, the root 0 came from an earlier step. Therefore, the roots are 0, 6. 41 and 1. 41.
![Factoring Sum or Difference of Cubes If you have a sum or difference of Factoring Sum or Difference of Cubes If you have a sum or difference of](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-8.jpg)
Factoring Sum or Difference of Cubes If you have a sum or difference of cubes such as a 3 + b 3 or a 3 – b 3, you can factor by using the following patterns. Note: The first and last term are cubed and these are binomials.
![Example Factor x 3 343 Note This is a binomial Are the first Example Factor x 3 + 343. Note: This is a binomial. Are the first](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-9.jpg)
Example Factor x 3 + 343. Note: This is a binomial. Are the first and last terms cubed? x 3 + 343 = (x)3 + (7)3 = ( x + 7 )( x 2 - 7 x + 49)
![Example Factor 64 a 4 27 a a64 a 3 27 Example Factor 64 a 4 – 27 a = a(64 a 3 – 27)](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-10.jpg)
Example Factor 64 a 4 – 27 a = a(64 a 3 – 27) Note: Binomial. Is the first and last terms cubes? = a( (4 a)3 – (3)3) Note: = a( 4 a - 3)( 16 a 2 + 12 a + 9 )
![Factor by Grouping Some four term polynomials can be factor by grouping Example Factor Factor by Grouping Some four term polynomials can be factor by grouping. Example. Factor](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-11.jpg)
Factor by Grouping Some four term polynomials can be factor by grouping. Example. Factor 3 x 3 + 7 x 2 +12 x + 28 Step 1 Pair the terms. Step 2 Factor out common factor from each pair. Identical factors Step 3 Factor out common factor from each term.
![Example Factor 3 x 3 7 x 2 12 x 28 Step Example Factor 3 x 3 + 7 x 2 -12 x - 28 Step](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-12.jpg)
Example Factor 3 x 3 + 7 x 2 -12 x - 28 Step 1 Note: Subtraction is the same as adding a negative Step 2 Step 3 Note: This factor can be further factored
![Solving Polynomial Equations Solve Set equation equal to zero Factor Set each factor equal Solving Polynomial Equations Solve Set equation equal to zero. Factor. Set each factor equal](https://slidetodoc.com/presentation_image_h/2f38b76a2dab27219085d947a7c517cf/image-13.jpg)
Solving Polynomial Equations Solve Set equation equal to zero. Factor. Set each factor equal to zero and solve.
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