Section 4.1
The Product, Quotient, and Power Rules for Exponents
OBJECTIVES
Multiply expressions using the product rule for exponents.
A
OBJECTIVES
Divide expressions using the quotient rule for exponents.
B
OBJECTIVES
Use the power rules to simplify expressions.
C
RULESSigns for Multiplication
1. When multiplying two numbers with the same sign, product is positive (+).
RULESSigns for Multiplication
2. When multiplying two numbers with different signs, product is negative (-).
RULESSigns for Division
1.When dividing two numbers with the same sign, product is positive (+).
RULESSigns for Division
2.When dividing two numbers with different signs, product is negative (-).
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:1. Product rule for exponents
xmxn = xm+n
Example:
x5•x6 = x5+6 = x11
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents
- > , 0=m m nn m n xx x
x
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents
Example:
p8
p3 = p8-3 = p5
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
3. Power rule for products
=k mk nkm n yy xx
RULES FOR EXPONENTSIf 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• •
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
4. Power rule for quotients
0=m m
m yx x y y
RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
4. Power rule for quotients Example:
6
= =3 3 6 184 4 6 24
a a ab b b
••
Section 4.1Exercise #1
Chapter 4Exponents and Polynomials
Find.
a. (2a3b)(– 6ab3 )
= (2 • – 6)a3+1 b1+3
= – 12a4b4
b. (– 2x 2yz)(– 6xy3z 4)
= ( – 2 • – 6)x 2 + 1 y1 + 3 z1 + 4
= 12x3y5z5
Find.
c. 18x5y7
– 9xy3
= 18
– 9
x5 – 1 y7 – 3
= – 2x 4y 4
Section 4.1Exercise #2
Chapter 4Exponents and Polynomials
Find.
3 2 3 3 3 3 2 3(2 ) = 2 x y x y
= 8x 9y6
b. ( – 3x 2y3 )2
a. (2x3y 2 )3
2 3 2 2 2 2 3 2( – 3 ) = ( – 3) x y x y
= 9x 4y6
Section 4.2
Integer Exponents
OBJECTIVES
Write an expression with negative exponents as an equivalent one with positive exponents.
A
OBJECTIVES
Write a fraction involving exponents as a number with a negative power.
B
OBJECTIVES
Multiply and divide expressions involving negative exponents.
C
RULESZero Exponent
0For 0, =1x x
– n 1= 0nx xx
If n is a positive integer,Negative Exponent
RULESnth Power of a Quotient
–1 =
nnxx
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
Section 4.2Exercise #4
Chapter 4Exponents and Polynomials
Simplify and write the answer without negative exponents.
– 71a. x
– 7– 1 = x
= x( – 1) ( – 7 )
= x 7
Simplify and write the answer without negative exponents.
b. x – 6
x – 6
= x – 6 – – 6
0 = = 1, 0xx
= x – 6 + 6
Section 4.2Exercise #5
Chapter 4Exponents and Polynomials
Simplify.– 3 4
2 3
– 2
2
b. 3
x yx 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
Simplify.
= 2 – 2 3 – 1( – 2) x – 5( – 2) y ( – 2)
= 2 – 2 3 2 x 10 y – 2
2 102 21 1 = 3
2 x
y
= 9x10
4y2
Section 4.3
Applicationof Exponents:Scientific Notation
OBJECTIVES
Write numbers in scientific notation.
A
OBJECTIVES
Multiply and divide numbers in scientific notation.
B
Solve applications.C
RULES
M10n
A number in scientific notation is written as
Where M is a number between 1 and 10 and n is an integer.
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
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.
PROCEDURE
3. Write (M10n).
(M10n)Writing a number in scientific notation
PROCEDUREMultiplying using scientific notation
1. Multiply decimal parts first. Write result in scientific notation.
PROCEDUREMultiplying using scientific notation
2. Multiply powers of 10 using product rule.
PROCEDUREMultiplying using scientific notation
3. Answer is product obtained in steps 1 and 2 after simplification.
Section 4.3Exercise #6
Chapter 4Exponents and Polynomials
a. 48,000,000
Write in scientific notation.
= 4 8000000 .
= 4.8107
b. 0.00000037
= 0.0000003 7
= 3.7 10 – 7
Section 4.3Exercise #7
Chapter 4Exponents and Polynomials
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
Section 4.4
Polynomials:An Introduction
OBJECTIVES
Classify polynomials.A
Find the degree of a polynomial.
B
OBJECTIVES
Write a polynomial in descending order.
C
Evaluate polynomials.D
DEFINITIONPolynomialAn algebraic expression formed using addition and subtraction on products of numbers and variables raised to whole number exponents.
Section 4.4Exercise #8
Chapter 4Exponents and Polynomials
Classify as a monomial (M), binomial (B), or trinomial (T).
a. 3x – 5
B, binomial
b. 5x3
M, monomial
c. 8x 2 – 2 + 5x
T, trinomial
Section 4.4Exercise #10
Chapter 4Exponents and Polynomials
Find the value.
– 16t 2 + 100 when t = 2
= – 16(2)2 + 100
= – 16(4) + 100
= – 64 + 100
= 36
Section 4.5
Addition and Subtraction of Polynomials
OBJECTIVES
Add polynomials.A
Subtract polynomials.B
OBJECTIVES
Find areas by adding polynomials.
C
Solve applications.D
Section 4.5Exercise #11
Chapter 4Exponents and Polynomials
Add.
2 – 4 + 8 – 3 + –5 – 4 + 2 2x x x x
= – 4x + 8x 2 – 3 – 5x 2 – 4 + 2x
= ( 8x 2 – 5x 2) + ( – 4x + 2x ) + ( – 3 – 4)
= 3x 2 – 2x – 7
Section 4.5Exercise #12
Chapter 4Exponents and Polynomials
23 – 2 – 5 – 2 + 82x x x x
= 3x 2 – 2x – 5x + 2 – 8x 2
= (3x 2 – 8x 2) + ( – 2x – 5x ) + 2
= – 5x 2 – 7x +2
Subtract 5x – 2 + 8x 2 from 3x2 – 2x.
Section 4.6
Multiplicationof Polynomials
OBJECTIVES
Multiply two monomials.A
Multiply a monomial and a binomial.
B
OBJECTIVES
Multiply two binomials using FOIL method.
C
Solve an application.D
PROCEDURE
First terms multiplied first.
FOIL Method for Multiplying Binomials
Outer terms multiplied second.
Inner terms multiplied third.
Last terms multiplied last.
Section 4.6Exercise #16
Chapter 4Exponents and Polynomials
Find (5x – 2y ) (4x – 3y ) .
= 20x 2 – 23xy + 6y 2
= 20x 2 – 15xy – 8xy + 6y 2F O I L
Section 4.7
Special Productof Polynomials
OBJECTIVES
Expand binomials of the form
A (X +A)2
B (X – A)2
C (X +A)(X – A)
OBJECTIVES
Multiply a binomial by a trinomial.
D
Multiply any two polynomials.
E
SPECIAL PRODUCTS
(X +A)(X +B)= X 2+(A+B)X +AB
SP1 or FOIL
SPECIAL PRODUCTS
SP2
(X +A)(X +A)=(X +A)2
= X 2+2AX +A2
SPECIAL PRODUCTS
SP3
(X -A)(X -A)=(X -A)2
= X 2 -2AX +A2
SPECIAL PRODUCTS
2 2( + )( - )= -X A X A X A
SP4
PROCEDUREMultiplying Any Two Polynomials (Term-By-Term Multiplication)
Multiply each term of one by every term of other and add results.
PROCEDUREAppropriate 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.
PROCEDUREAppropriate Method for Multiplying Two Polynomials:2. Are the two binomials in the
product the sum and difference of the same two terms?
PROCEDUREAppropriate Method for Multiplying Two Polynomials:
Answer has two terms.
If so, use SP4.
PROCEDUREAppropriate Method for Multiplying Two Polynomials:
3. Is the binomial product different from previous two?
Answer has three or four terms.If so, use FOIL.
PROCEDUREAppropriate 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.
Section 4.7Exercise #18
Chapter 4Exponents and Polynomials
Expand.
(2x – 7y )2 (a – b)
2 = a
2– 2 ab + b
2
= 4x 2 – 28xy + 49y 2
= (2x)2
– 2 (2x)(7y) + ( 7y ) 2
Section 4.7Exercise #19
Chapter 4Exponents and Polynomials
Find (2x – 5y )(2x + 5y).
= (2x )2 – (5y )2
= 4x 2 – 25y 2
Section 4.7Exercise #20
Chapter 4Exponents and Polynomials
Find (x + 2)(x2 + 5x + 3)
= x (x2 + 5x + 3) + 2(x2 + 5x + 3)
= x 3 + 5x 2 + 3x + 2x 2 + 10x + 6
= x 3 + (5x 2 + 2x 2 ) + (3x + 10x ) + 6
= x 3 + 7x 2 + 13x + 6
Section 4.8
Divisionof Polynomials
OBJECTIVES
Divide a polynomial by a monomial.
A
Divide one polynomial by another polynomial.
B
RULETo Divide A Polynomial By A Monomial
Divide each term in polynomial by monomial.
Section 4.8Exercise #25
Chapter 4Exponents and Polynomials
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