Math 40
Chapter 4
Lecture Notes
Professor Miguel Ornelas
1
M. Ornelas
Math 40 Lecture Notes
Section 4.1
Multiplication With Exponents
Product Rule for Exponents
am · an = am+n
Simplify.
1.
22 · 2 6
2. x7 x4
3. y · y 3 · y 5
4.
43 · 44
5.
(3x6 )(5x)
6.
(−2x3 p2 )(4xp10 )
7.
(2x5 )3 (3x4 )
8.
(5a3 )2 (2a7 )
2.
4,000,000
Scientific Notation
Write each number in scientific notation.
1.
730,000
Section 4.1 continued on next page. . .
2
Section 4.1
M. Ornelas
Math 40 Lecture Notes
Write each number in standard notation.
1.
7.7 × 108
2.
1.025 × 103
Simplify the expression and write answer in scientific notation.
(4 × 1010 )(3 × 103 )
Section 4.2
Division With Exponents
Zero Exponent
If a does not equal 0, then a0 = 1.
Simplify.
1.
70
2. −70
3.
(2x + 5)0
4.
Quotient Rule for Exponents
am
= am−n
an
Section 4.2 continued on next page. . .
3
2x0
Section 4.1 (continued)
M. Ornelas
Math 40 Lecture Notes
Simplify.
1.
x7
x4
2.
58
52
3.
20x6
4x5
4.
12y 10 z 7
14y 8 z 7
2.
(−4)−4
4.
m5
m15
6.
1
t−5
Negative Exponents
a−n =
1
an
Simplify.
1.
5−2
−3
3.
2x
5.
2−1 + 3−2
Section 4.2 continued on next page. . .
4
Section 4.2 (continued)
M. Ornelas
Math 40 Lecture Notes
7.
3x3 y 2 z
27xy 2 z 3
8.
9.
2x−7 y 2
10xy −5
10.
2−3
2−1
(3x−3 )(x2 )
x6
Power Rule, Power of a Product and Quotient Rules for Exponents
1. (am )n = am·n
2. (ab)m = am bm
3.
a n
b
=
an
bn
Simplify.
1.
(x5 )7
3.
3p4
q5
2.
(5x2 )3
4.
(5x−6 y 2 )2
(x5 y −6 )−3
2
Section 4.2 continued on next page. . .
5
Section 4.2 (continued)
M. Ornelas
Math 40 Lecture Notes
Scientific Notation
Write each number in scientific notation.
1.
0.000045
2.
0.00000105
2.
5 × 10−3
Write each number in standard notation.
1.
3.08 × 10−6
Simplify the expression and write answer in scientific notation.
1.
2.5 × 10−6
5 × 10−4
2.
(8 × 104 )(5 × 1010 )
2 × 10−7
Section 4.3
Addition and Subtraction of Polynomials
Definitions
1. A polynomial is a finite sum of terms.
2. The degree of a polynomial is the largest degree of all its terms.
Section 4.3 continued on next page. . .
6
Section 4.2 (continued)
M. Ornelas
Math 40 Lecture Notes
Section 4.3 (continued)
Find the degree of each polynomial and indicate whether the polynomial is also a monomial, binomial, or
trinomial.
1.
3
7x3 − x + 2
4
2. n5 − 4n3 + 2n − 7
3.
9a2 + 2
4.
8x5 + 5x3 + 1
Evaluating a Polynomial
Evaluate the polynomial 3x2 − 2x − 5 for x = −3.
Combining Like Terms
Add or Subtract.
1.
(7y 3 − y 2 − y + 11) + (5y 3 − 4y − 6)
2.
8x4 − 5x + 3x4 + 2x
4
3
4
3
4.
3.
(13a − 7a − 9) − (−2a + 8a − 12)
5.
Subtract 5x2 y 2 − 3xy 2 + 5y 3 from 11x2 y 2 − 7xy 2
Section 4.3 continued on next page. . .
7
1
5 3 1
x − x−
8
4
3
1 3 1
1
− − x + x−
2
4
3
M. Ornelas
Math 40 Lecture Notes
Section 4.4
Multiplication of Polynomials
Multiply.
1.
(3x4 )(2x2 )
3. −3x2 (4x2 − 6x + 1)
2.
(−5m3 np2 )(−8mnp5 )
4.
−x2 y(7x2 y + 3xy − 11)
(2x − 3)(x2 − 6x + 7)
Multiply and simplify the product if possible.
1.
(x + 3)(2x + 5)
2.
3.
(2x − 7)(3x − 4)
4.
5.
(3a4 + 2)(2a2 + 5)
Section 4.4 continued on next page. . .
6.
8
2x −
1
2
3
x+
2
(x + 4)(x − 5) + (−5)(2)
Section 4.3 (continued)
M. Ornelas
Math 40 Lecture Notes
Section 4.5
Special Products
Square of a Binomial
(a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
Product of the Sum and a Difference
(a + b)(a − b) = a2 − b2
Multiply.
1.
2.
(x + 5)2
2
2
3.
(4m − 3n)
5.
(7y 5 + 6)(7y 5 − 6)
Section 4.5 continued on next page. . .
(4y + 1)2
4.
6.
9
1
t +
5
2
1
2
t −
5
(x + 5)2 − (x − 5)2
Section 4.4 (continued)
M. Ornelas
Math 40 Lecture Notes
Section 4.5 (continued)
Section 4.6
Division of Polynomials
Dividing a Polynomial by a Monomial
1.
Divide 18a3 − 12a2 + 30a by 6a
2.
3x5 y 2 − 15x3 y − x2 y − 6x
x2 y
Dividing by a Polynomial
1.
Divide 2x2 − x − 10 by x + 2
2.
(6x2 − 19x + 12) ÷ (3x − 5)
3.
(5x3 + 9x2 − 10x + 30) ÷ (x + 3)
4.
Divide x3 + 4x + 5 by x + 1
10