Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long...
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Transcript of Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long...
![Page 1: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/1.jpg)
Polynomial and Synthetic Division
#3
![Page 2: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/2.jpg)
Now let’s look at another method to divide… Why??? Sometimes it is easier…
![Page 3: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/3.jpg)
Synthetic Division
Synthetic Division is a ‘shortcut’ for polynomial division that only works when dividing by a linear factor (x + b).
It involves the coefficients of the dividend, and the zero of the divisor.
![Page 4: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/4.jpg)
Synthetic DivisionSynthetic Division
The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.)
most
vertical pattern: ADD termsDiagonal pattern: MULTIPLY terms
![Page 5: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/5.jpg)
Example
Divide: Step 1:
Write the coefficients of the dividend in a upside-down division symbol.
1 5 6
1
652
x
xx
![Page 6: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/6.jpg)
Example Step 2:
Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0 , so the zero is 1.
1 5 61
1
652
x
xx
![Page 7: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/7.jpg)
Example
Step 3:Carry down the first coefficient.
1 5 61
1
1
652
x
xx
![Page 8: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/8.jpg)
Example
Step 4:Multiply the zero by this number. Write the
product under the next coefficient.
1 5 61
1
1
1
652
x
xx
![Page 9: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/9.jpg)
Example
Step 5:Add.
1 5 61
1
1
6
1
652
x
xx
![Page 10: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/10.jpg)
Example
Step 6 etc.:Repeat as necessary
1 5 61
1
1
6
6
12
1
652
x
xx
![Page 11: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/11.jpg)
step 7
The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original.
1 5 61
1
1
6
6
12 1
126
x
x
1
652
x
xx
![Page 12: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/12.jpg)
Using Synthetic Division
Use synthetic division to divide x4 – 10x2 – 2x + 4 by x + 3.
Solution:
You should set up the array as follows. Note that a zero is included for the missing +x3 term in the dividend.
![Page 13: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/13.jpg)
Example
Divide: Step 1:
Write the coefficients of the dividend in a upside-down division symbol.
2 3 4
2x2 + 3x + 4 x - 1
![Page 14: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/14.jpg)
Example Step 2:
Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0 , so the zero is 1.
2 3 41
2x2 + 3x + 4 x - 1
![Page 15: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/15.jpg)
Example
Step 3:Carry down the first coefficient.
2 3 41
2
2x2 + 3x + 4 x - 1
![Page 16: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/16.jpg)
Example
Step 4:Multiply the zero by this number. Write the
product under the next coefficient.
2 3 41
2
2
2x2 + 3x + 4 x - 1
![Page 17: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/17.jpg)
Example
Step 5:Add.
2 3 41
2
2
5
2x2 + 3x + 4 x - 1
![Page 18: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/18.jpg)
Example
Step 6 etc.:Repeat as necessary
2 3 41
2
2
5
5
9
2x2 + 3x + 4 x - 1
![Page 19: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/19.jpg)
step 7
The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original.
2 3 41
2
2
5
5
9
1
652
x
xx
![Page 20: Polynomial and Synthetic Division #3. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022042703/5697bfea1a28abf838cb73cb/html5/thumbnails/20.jpg)
****Lab Application, practice