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[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§6.4 Divide§6.4 DividePolyNomialsPolyNomials
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §6.3 → Complex Rational
Expressions
Any QUESTIONS About HomeWork• §6.3 → HW-25
6.3 MTH 55
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt3
Bruce Mayer, PE Chabot College Mathematics
§6.4 Polynomial Division §6.4 Polynomial Division
Dividing by a Monomial
Dividing by a BiNomial
Long Division
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt4
Bruce Mayer, PE Chabot College Mathematics
Dividing by a MonomialDividing by a Monomial
To divide a polynomial by a monomial, divide each term by the monomial.
EXAMPLE – Divide: x5 + 24x4 − 12x3 by 6x
Solution5 4 3 5 4 324 12 24 12
6 6 6 6
x x x x x x
x x x x
5 1 4 1 3 11 24 12
6 6 6x x x
4 3 214 2
6x x x
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt5
Bruce Mayer, PE Chabot College Mathematics
Example Example Monomial Division Monomial Division
Divide: 5 4 3 2 2 221 14 7 7a b a b a b a b
Solution: 5 4 3 2 2 5 4 3 2 2
2 2 2 2
21 14 7 21 14 7
7 7 7 7
a b a b a b a b a b a b
a b a b a b a b
5 2 4 1 13 2221 14 7
7 7 7a b a b
123 33 abba
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt6
Bruce Mayer, PE Chabot College Mathematics
Dividing by a BinomialDividing by a Binomial
For divisors with more than one term, we use long division, much as we do in arithmetic.
Polynomials are written in descending order and any missing terms in the dividend are written in, using 0 (zero) for the coefficients.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt7
Bruce Mayer, PE Chabot College Mathematics
Recall Arithmetic Long DivisionRecall Arithmetic Long Division
Recall Whole-No. Long Division
Divide: 157
12 157
12
37
36
1
13
1
2
Divisor Quotient
Remainder
Quotient 13
Divisor 12
Remainder 1
= Dividend= 157
•++
•
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt8
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Divide 2x³ + 3x² - x + 1 by x + 2
3 22 2 3 1x x x x x + 2 is the divisor
The quotient will be here.
2x³ + 3x² - x + 1 is the dividend
Use an IDENTICAL Long Division process when dividing by BiNomials or Larger PolyNomials; e.g.;
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt9
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
First divide the first term of the dividend, 2x³, by x (the first term of the divisor).
3 22 2 3 1x x x x
22xThis gives 2x². This will be the first term of the quotient.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt10
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Now multiply (x+2) by 2x²
3 22 2 3 1x x x x 3 22 4x x
22x
2xand subtract
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt11
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Bring down the next term, -x.
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt12
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Now divide –x², the first term of –x² - x, by x, the first term of the divisor
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
which gives –x.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt13
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Multiply (x +2) by -x
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
xand subtract
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt14
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Bring down the next term, 1 x
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
1
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt15
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Divide x, the first term of x + 1, by x, the first term of the divisor
13 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x 1which gives 1
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt16
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
Multiply x + 2 by 1
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x
1
12x 1and subtract
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt17
Bruce Mayer, PE Chabot College Mathematics
Binomial Div. Binomial Div. Step by Step Step by Step
The remainder is –1.
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x
1
12x 1
The quotient is 2x² - x + 1
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt18
Bruce Mayer, PE Chabot College Mathematics
Example Example BiNomial Division BiNomial Division
Divide x2 + 7x + 12 by x + 3.
Solution
2
2
3 7 12
( )
3
4
x
x
x x x
x
x
Subtract by changing signs and adding
Multiply (x + 3) by x, using the distributive law
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt19
Bruce Mayer, PE Chabot College Mathematics
Example Example BiNomial Division BiNomial Division
Solution – Cont.
2
2
4
+ 3 7 12
( 3 )
12
( )
2
0
4
1
4
x
x
xx x x
x x
Subtract
Multiply 4 by the divisor, x + 3, using the distributive law
Bring Down the +12
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt20
Bruce Mayer, PE Chabot College Mathematics
Example Example BiNomial Division BiNomial Division
Divide 15x2 − 22x + 14 by (3x − 2) Solution
2
2
3 2 15 22 14
( )
12 14
( 12 8)
6
15 1
5 4
0
x
x
x
x
x
x
x x
The answer is 5x − 4 with R6. We can also write the answer as:
65 4 .
3 2x
x
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt21
Bruce Mayer, PE Chabot College Mathematics
Example Example BiNomial Division BiNomial Division
Divide x5 − 3x4 − 4x2 + 10x by (x − 3) Solution
The Result
5 4 3 2
5 4
3 2
2
3 3 0 4 10 0
( 3 )
0 4 10
4 12
2 0
( 2 6)
x x x x x x
x x
x x x
x x
x
x
4 4 2
6
x x
4 6
4 2 .3
x xx
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt22
Bruce Mayer, PE Chabot College Mathematics
Divide 3Divide 3xx22 −− 4 4xx −− 15 by 15 by xx −− 3 3
SOLUTION: Place the TriNomial under the Long Division Sign and start the Reduction Process
23 3 4 15x x x 3x
2(3 )9x x
5 15x
Divide 3x2 by x: 3x2/x = 3x.
Multiply x – 3 by 3x.
Subtract by mentally changing signs and adding −4x + 9x = 5x.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt23
Bruce Mayer, PE Chabot College Mathematics
Divide 3Divide 3xx22 −− 4 4xx −− 15 by 15 by xx −− 3 3 SOLUTION: next divide the leading
term of this remainder, 5x, by the leading term of the divisor, x.
23 3 4 15x x x 3 5x
2(3 9 )x x
5 15x
Divide 5x by x: 5x/x = 5.
Multiply x – 3 by 5.
Subtract. Our remainder is now 0.
5(5 1 )x
0
CHECK: (x − 3)(3x + 5) = 3x2 − 4x − 15 The quotient is 3x + 5.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt24
Bruce Mayer, PE Chabot College Mathematics
Formal Division AlgorithmFormal Division Algorithm
If a polynomial F(x) is divided by a polynomial D(x), with D(x) ≠ 0, there are unique polynomials Q(x) and R(x) such thatF(x) = D(x) • Q(x) + R(x)
Dividend Divisor Quotient Remainder
Either R(x) is the zero polynomial, or the degree of R(x) is LESS than the degree of D(x).
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt25
Bruce Mayer, PE Chabot College Mathematics
PolyNomial Long DivisionPolyNomial Long Division
1. Write the terms in the dividend and the divisor in descending powers of the variable.
2. Insert terms with zero coefficients in the dividend for any missing powers of the variable
3. Divide the first terms in the dividend by the first terms in the divisor to obtain the first term in the quotient.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt26
Bruce Mayer, PE Chabot College Mathematics
PolyNomial Long DivisionPolyNomial Long Division
4. Multiply the divisor by the first term in the quotient, and subtract the product from the dividend.
5. Treat the remainder obtained in Step 4 as a new dividend, and repeat Steps 3 and 4. Continue this process until a remainder is obtained that is of lower degree than the divisor.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt27
Bruce Mayer, PE Chabot College Mathematics
Example Example TriNomial Division TriNomial Division
Divide x4 13x2 x 35 by x2 x 6.
SOLN
x2 x 6 x4 0x3 13x2 x 35
x4 x3 6x2
x3 7x2 x 35
x3 x2 6x
6x2 7x 35
x2 x
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt28
Bruce Mayer, PE Chabot College Mathematics
Example Example TriNomial Division TriNomial Division
SOLNcont. x2 x 6 x4 0x3 13x2 x 35
x4 x3 6x2
x3 7x2 x 35
x3 x2 6x
6x2 7x 35
6x2 6x 36
x 1
x2 x 6
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt29
Bruce Mayer, PE Chabot College Mathematics
Example Example TriNomial Division TriNomial Division
Divide x4 13x2 x 35 by x2 x 6.
The Quotient = x2 x 6.
The Remainder = x 1.
Write the Result in Concise form:
x4 13x2 x 35
x2 x 6x2 x 6
x 1
x2 x 6.
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt30
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §6.4 Exercise Set• 30, 32, 40
BiNomialDivision
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt31
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
PolynomialDivisionin base2
From UC Berkeley Electrical-Engineering 122 Course
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt32
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt33
Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
[email protected] • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt34
Bruce Mayer, PE Chabot College Mathematics
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
xy