the product, quotient, and power rules for exponents...quotient rule for exponents example: p 8 p3 =...
TRANSCRIPT
![Page 1: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/1.jpg)
The Product,
Quotient, and Power
Rules for Exponents
![Page 2: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/2.jpg)
OBJECTIVES
Multiply expressions
using the product rule
for exponents.
A
![Page 3: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/3.jpg)
OBJECTIVES
Divide expressions
using the quotient rule
for exponents.
B
![Page 4: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/4.jpg)
OBJECTIVES
Use the power rules to
simplify expressions.
C
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RULES
Signs for Multiplication
1. When multiplying two
numbers with the same
sign, product is positive (+).
![Page 6: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/6.jpg)
RULES
Signs for Multiplication
2. When multiplying two
numbers with different signs,
product is negative (-).
![Page 7: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/7.jpg)
RULES
Signs for Division
1.When dividing two
numbers with the same
sign, product is positive (+).
![Page 8: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/8.jpg)
RULES
Signs for Division
2.When dividing two numbers
with different signs, product
is negative (-).
![Page 9: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/9.jpg)
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
1. Product rule for exponents
xmxn = xm+n
Example:
x5•x6 = x5+6 = x11
![Page 10: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/10.jpg)
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
2. Quotient rule for exponents
- > , 0=m m nn m n xx x
x
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RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
2. Quotient rule for exponents
Example:
p8
p3= p8-3 = p5
![Page 12: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/12.jpg)
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
3. Power rule for products
=k
mk nkm n yy xx
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RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
3. Power rule for products
Example:
= =4
4 3 4 4 3 4 16 12x y x xy y• •
![Page 14: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/14.jpg)
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
4. Power rule for quotients
0=m m
myx x
y y
![Page 15: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/15.jpg)
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
4. Power rule for quotients
Example: 6
= =3 3 6 18
4 4 6 24a a a
b b b
•
•
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Section 4.1
Exercise #1
Chapter 4
Exponents and Polynomials
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Find.
a. (2a3b)(– 6ab3 )
= (2 • – 6)a3+1 b1+3
= – 12a4b4
b. (– 2x2yz)(– 6xy3z 4)
= ( – 2 • – 6)x2 + 1 y1 + 3 z1 + 4
= 12x3y5z5
![Page 18: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/18.jpg)
Find.
c. 18x5y7
– 9xy3
=
18
– 9
x5 – 1 y7 – 3
= – 2x4y4
![Page 19: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/19.jpg)
Section 4.1
Exercise #2
Chapter 4
Exponents and Polynomials
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Find.
3 2 3 3 3 3 2 3(2 ) = 2x y x y
= 8x9y6
b. ( – 3x2y3 )2
a. (2x3y2 )3
2 3 2 2 2 2 3 2( – 3 ) = ( – 3) x y x y
= 9x4y6
![Page 21: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/21.jpg)
Section 4.2
Integer Exponents
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OBJECTIVES
Write an expression with
negative exponents as
an equivalent one with
positive exponents.
A
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OBJECTIVES
Write a fraction
involving exponents as
a number with a
negative power.
B
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OBJECTIVES
Multiply and divide
expressions involving
negative exponents.
C
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RULES
Zero Exponent 0For 0, =1x x
– n 1= 0nx xx
If n is a positive integer,
Negative Exponent
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RULES
nth Power of a Quotient
–1 =
nnx
x
![Page 27: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/27.jpg)
RULES
x–m
y–n = yn
xm
For any nonzero numbers
x and y and any positive
integers m and n:
Simplifying Fractions with
Negative Exponents
![Page 28: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/28.jpg)
Section 4.2
Exercise #4
Chapter 4
Exponents and Polynomials
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Simplify and write the answer without negative
exponents.
– 7
1a. x
– 7– 1 = x
= x( – 1) ( – 7 )
= x 7
![Page 30: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/30.jpg)
Simplify and write the answer without negative
exponents.
b. x – 6
x – 6
= x – 6 – – 6
0 = = 1, 0xx
= x – 6 + 6
![Page 31: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/31.jpg)
Section 4.2
Exercise #5
Chapter 4
Exponents and Polynomials
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Simplify.
– 3 4
2 3
– 2
2
b. 3
x y
x y
= 2 –2 x – 3 –2
y 4 –2
3–2 x
2 –2 y
3 –2
= 2 –2 x 6 y –8
3–2 x
– 4 y
–6
= 32 x 6 – – 4
y –8 –(–6)
22
=
9 x10 y –2
4
= 9 x10
4y2
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Simplify.
= 2 – 2 3 – 1( – 2) x – 5( – 2) y ( – 2)
= 2 – 2 3 2 x 10 y – 2
2 102 21 1 = 3
2x
y
= 9x10
4y2
![Page 34: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/34.jpg)
Section 4.3
Application
of Exponents:
Scientific Notation
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OBJECTIVES
Write numbers in
scientific notation.
A
![Page 36: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/36.jpg)
OBJECTIVES
Multiply and divide
numbers in scientific
notation.
B
Solve applications. C
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RULES
M10n
A number in scientific notation
is written as
Where M is a number between
1 and 10 and n is an integer.
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PROCEDURE
1. Move decimal point in
number so there is only
one nonzero digit to its
left.
(M10n)
The resulting number is M.
Writing a number in scientific
notation
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PROCEDURE
2. If the decimal point is moved
to the left, n is positive;
(M10n)
Writing a number in scientific
notation
If the decimal point is moved
to the right, n is negative.
![Page 40: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/40.jpg)
PROCEDURE
3. Write (M10n).
(M10n)
Writing a number in scientific
notation
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PROCEDURE
Multiplying using scientific
notation
1. Multiply decimal parts first.
Write result in scientific
notation.
![Page 42: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/42.jpg)
PROCEDURE
Multiplying using scientific
notation
2. Multiply powers of 10
using product rule.
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PROCEDURE
Multiplying using scientific
notation
3. Answer is product
obtained in steps 1 and 2
after simplification.
![Page 44: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/44.jpg)
Section 4.3
Exercise #6
Chapter 4
Exponents and Polynomials
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a. 48,000,000
Write in scientific notation.
= 4 8000000 .
= 4.8107
b. 0.00000037
= 0.0000003 7
= 3.7 10 – 7
![Page 46: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/46.jpg)
Section 4.3
Exercise #7
Chapter 4
Exponents and Polynomials
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Perform the indicated operations.
4 6a. 3 10 7.1 10
4 + 6 = 3 7.1 10
= 21.3 1010
= 2.13 101 + 10
= 2.13 1011
= 2.13 101 1010
![Page 48: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/48.jpg)
Section 4.4
Polynomials:
An Introduction
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OBJECTIVES
Classify polynomials. A
Find the degree of a
polynomial.
B
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OBJECTIVES
Write a polynomial in
descending order.
C
Evaluate polynomials. D
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DEFINITION
Polynomial
An algebraic expression
formed using addition and
subtraction on products of
numbers and variables raised
to whole number exponents.
![Page 52: The Product, Quotient, and Power Rules for Exponents...Quotient rule for exponents Example: p 8 p3 = p8-3 = p5 RULES FOR EXPONENTS If m, n, and k are positive integers, then: 3. Power](https://reader035.vdocument.in/reader035/viewer/2022081507/5f07a1887e708231d41df2b1/html5/thumbnails/52.jpg)
Section 4.4
Exercise #8
Chapter 4
Exponents and Polynomials
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Classify as a monomial (M), binomial (B), or trinomial (T).
a. 3x – 5
B, binomial
b. 5x3
M, monomial
c. 8x2 – 2 + 5x
T, trinomial
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Section 4.4
Exercise #10
Chapter 4
Exponents and Polynomials
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Find the value.
– 16t2 + 100 when t = 2
= – 16(2)2 + 100
= – 16(4) + 100
= – 64 + 100
= 36
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Section 4.5
Addition and Subtraction
of Polynomials
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OBJECTIVES
Add polynomials. A
Subtract polynomials. B
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OBJECTIVES
Find areas by adding
polynomials.
C
Solve applications. D
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Section 4.5
Exercise #11
Chapter 4
Exponents and Polynomials
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Add.
2 – 4 + 8 – 3 + –5 – 4 + 2 2x x x x
= – 4x + 8x2 – 3 – 5x2 – 4 + 2x
= ( 8x2 – 5x2) + ( – 4x + 2x) + ( – 3 – 4)
= 3x2 – 2x – 7
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Section 4.5
Exercise #12
Chapter 4
Exponents and Polynomials
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23 – 2 – 5 – 2 + 82x x x x
= 3x2 – 2x – 5x + 2 – 8x2
= (3x2 – 8x2) + ( – 2x – 5x ) +2
= – 5x2 – 7x +2
Subtract 5x – 2 + 8x2 from 3x2 – 2x.
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Section 4.6
Multiplication
of Polynomials
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OBJECTIVES
Multiply two monomials. A
Multiply a monomial and
a binomial.
B
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OBJECTIVES
Multiply two binomials
using FOIL method.
C
Solve an application. D
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PROCEDURE
First terms multiplied first.
FOIL Method for Multiplying
Binomials
Outer terms multiplied second.
Inner terms multiplied third.
Last terms multiplied last.
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Section 4.6
Exercise #16
Chapter 4
Exponents and Polynomials
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Find (5x – 2y) (4x – 3y) .
= 20x2 – 23xy + 6y2
= 20x2 – 15xy – 8xy + 6y2F O I L
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Section 4.7
Special Product
of Polynomials
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OBJECTIVES
Expand binomials of the form
A (X +A)2
B (X – A)2
C (X +A)(X – A)
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OBJECTIVES
Multiply a binomial by a
trinomial.
D
Multiply any two
polynomials.
E
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SPECIAL PRODUCTS
(X +A)(X +B)= X 2+(A+B)X +AB
SP1 or FOIL
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SPECIAL PRODUCTS
SP2
(X +A)(X +A)=(X +A)2
= X 2+2AX +A2
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SPECIAL PRODUCTS
SP3
(X -A)(X -A)=(X -A)2
= X 2 -2AX +A2
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SPECIAL PRODUCTS
2 2( + )( - )= -X A X A X A
SP4
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PROCEDURE
Multiplying Any Two
Polynomials
(Term-By-Term Multiplication)
Multiply each term of one by
every term of other and add
results.
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PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
1. Is the product the square
of a binomial?
Both answers have three terms.
If so, use SP2 or SP3.
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PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
2. Are the two binomials in the
product the sum and
difference of the same two
terms?
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PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
Answer has two terms.
If so, use SP4.
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PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
3. Is the binomial product
different from previous
two?
Answer has three or four terms. If so, use FOIL.
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PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
4. Is product still different?
If so, multiply every term
of first polynomial by
every term of second and
collect like terms.
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Section 4.7
Exercise #18
Chapter 4
Exponents and Polynomials
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Expand.
(2x – 7y)2
(a – b)2
= a2– 2 ab + b
2
= 4x2 – 28xy + 49y2
= (2x)2– 2 (2x)(7y) + ( 7y ) 2
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Section 4.7
Exercise #19
Chapter 4
Exponents and Polynomials
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Find (2x – 5y)(2x + 5y).
= (2x)2 – (5y)2
= 4x2 – 25y2
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Section 4.7
Exercise #20
Chapter 4
Exponents and Polynomials
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Find (x + 2)(x2 + 5x + 3)
= x(x2 + 5x + 3) + 2(x2 + 5x + 3)
= x 3 + 5x2 + 3x + 2x2 + 10x + 6
= x 3 + (5x2 + 2x2 ) + (3x + 10x) + 6
= x 3 + 7x2 + 13x + 6
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Section 4.8
Division
of Polynomials
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OBJECTIVES
Divide a polynomial by a
monomial.
A
Divide one polynomial
by another polynomial.
B
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RULE
To Divide A Polynomial By A
Monomial
Divide each term in
polynomial by monomial.
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Section 4.8
Exercise #25
Chapter 4
Exponents and Polynomials
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x – 2 2x3 + 0x 2 – 9x + 5
2x3 – 4x 2
4x 2 – 9x + 5
4x 2 – 8x
– 1x + 5
– 1x + 2
3
2x 2 + 4x – 1 R 3
Divide.
2x3 – 9x + 5 by x – 2
Remainder