Polynomial and Synthetic Division

1
Polynomial and Synthetic Division

• 3

2
Common Core objectives

• Students will be able to use long division to
  divide polynomials by other polynomials. Use
  synthetic division to divide polynomials by
  binomials of the form (x k). Use the Remainder
  Theorem and the Factor Theorem.
• Teacher will instruct students to learn about
  the process of division. Learn about a couple of
  theorems to help in factoring and solving higher
  level polynomials

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Now lets look at another method to divide

• Why???
• Sometimes it is easier

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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.

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Synthetic 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 terms
Diagonal pattern MULTIPLY terms
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Example

• Divide
• Step 1
• Write the coefficients of the dividend in a
  upside-down division symbol.

1
5
6
7
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.

5
6
1
1
8
Example

• Step 3
• Carry down the first coefficient.

1
5
6
1
1
9
Example

• Step 4
• Multiply the zero by this number. Write the
  product under the next coefficient.

1
5
6
1
1
1
10
Example

• Step 5
• Add.

1
5
6
1
1
1
6
11
Example

• Step 6 etc.
• Repeat as necessary

1
5
6
1
1
6
12
1
6
12
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
6
1
1
6
12
1
6
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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.

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Example
2x2 3x 4 x - 1

• Divide
• Step 1
• Write the coefficients of the dividend in a
  upside-down division symbol.

2
3
4
15
Example
2x2 3x 4 x - 1

• Step 2
• Take the zero (or root) from the divisor, and
  write it on the left, x 1 0 , so the zero is
  1.

3
4
2
1
16
Example
2x2 3x 4 x - 1

• Step 3
• Carry down the first coefficient.

2
3
4
1
2
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Example
2x2 3x 4 x - 1

• Step 4
• Multiply the zero by this number. Write the
  product under the next coefficient.

2
3
4
1
2
2
18
Example
2x2 3x 4 x - 1

• Step 5
• Add.

2
3
4
1
2
2
5
19
Example
2x2 3x 4 x - 1

• Step 6 etc.
• Repeat as necessary

2
3
4
1
2
5
9
2
5
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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
4
1
2
5
9
2
5
21
Lab Application, practice