Post on 17-Dec-2015
transcript
2
Objectives
• I can use long division to divide two polynomials
• I can use synthetic division to divide a polynomial by a binomial (x – r)
3
Dividing Numbers
4164
When you divide a number by another number and there is no remainder:
Then the divisor is a factor!!
Also the quotient becomes another factor!!!
Dividend
Divisor
Quotient
4
Dividing Polynomials
Long division of polynomials is similar to long division of whole numbers.
dividend = (quotient • divisor) + remainder
The result is written in the form:
quotient +divisor
remainder divisor dividend
When you divide two polynomials you can check the answer using the following:
5
+ 2 2 3 1 2 xxx
Example: Divide x2 + 3x – 2 by x + 1 and check the answer.
x
x2 + x2x – 22x + 2
– 4
remainder
xx
xxx
22 1.
xxxx 2)1(2.
xxxxx 2)()3( 22 3.
22
2 x
xxx4.
22)1(2 xx5.
4)22()22( xx6.
Answer: x + 2 +1x
– 4
Dividing Polynomials
6
Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer.
1 4 0 2 2 2 23 xxxx Write the terms of the dividend in
descending order.
23
2
2x
x
x1.
x2
232 22)22( xxxx 2.
2x3 – 2x2
2233 2)22(2 xxxx 3.
2x2 + 4x
xx
x
2
2 2
4.
+ x
xxxx 22)22( 2 5.
2x2 – 2x
xxxxx 6)22()42( 22 6.
6x – 1
32
6
x
x7.
+ 3
66)22(3 xx8.
6x – 6
remainder5)66()16( xx9.
5
Answer: x2 + x + 322
x5
Since there is no x2 term in the
dividend, add 0x2 as a placeholder.
7
6 5 2 2 xxxx
x2 – 2x
– 3x + 6
– 3
– 3x + 60
Answer: x – 3 with no remainder.
Example: Divide x2 – 5x + 6 by x – 2.
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Dividing by Synthetic Division
• Synthetic Division is a method to divide any polynomial by a binomial.
• The steps must be followed exactly in order or you will not get the correct end result
• The following slide shows the steps for one complete problem.
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Find: (6x3- 19x2 + x + 6) (x-3)
• Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term.
• Bring down the coefficients from the dividend and make a row.
• Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line.
• Bring down 1st coefficient under the line. Multiply it by “r” and add to next column. Then repeat.
• New row of numbers are the coefficients of the quotient starting with one power less.
• 6x3 – 19x2 + 1x + 6
6 -19 1 6
3
6
18
-1
-3
-2
-6
0
6x2 – 1x – 2 (No remainder)
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Find: (4x4- 5x2 + 2x + 4) (x+1)
• Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term.
• Bring down the coefficients from the dividend and make a row.
• Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line.
• Bring down 1st coefficient under the line. Multiply it by “r” and add to next column. Then repeat.
• New row of numbers are the coefficients of the quotient starting with one power less.
• 4x4 + 0x3 – 5x2 + 2x + 4
4 0 -5 2 4
-1
4
-4
-4
4
-1
1
3
-3
1
1
13144 23
xxxx
11
16
Synthetic division is a shorter method of dividing polynomials.
This method can be used only when the divisor is of the form
x – a. It uses the coefficients of each term in the dividend.
Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division.
3 2 – 12
Since the divisor is x – 2, a = 2.
3
1. Bring down 3
2. (2 • 3) = 6
6
8 15
3. (2 + 6) = 8
4. (2 • 8) = 16
5. (–1 + 16) = 15coefficients of quotient remainder
value of a coefficients of the dividend
3x + 8Answer: 2
x15