Notes for Week 2
MAT 0024C
Name: ______________________________
2.4 Applying the Principles to Formulas

Isolating a variable
Examples:
Q: Solve for a: x  a  3 y .
A:
Q: Solve for w: 19  2l  2w .
A:
Q: Solve for a: 5(2n  a )  bm  c .
A:
Q: Solve for w:
w x
  y.
4 6
A:
Q: Solve for v: D 
m
.
v
A:
1
2.5 Translating Word Sentences to Equations
Examples:
Q: Translate the sentence to an equation and then solve: Six added to p is
negative two.
A:
Q: Translate the sentence to an equation and then solve: Three sevenths of a
number is equal to negative nine-eighths.
A:
Q: Translate the sentence to an equation and then solve: Tripling the difference
of a number and five produces negative fifteen.
A:
Q: Translate the sentence to an equation and then solve: Five more than four
times a number is equal to seven subtracted from that number.
A:
2
2.6 Solving Linear Inequalities
*The only difference between solving an inequality and an equality is that if you
multiply or divide by a negative number, reverse the inequality.
*Symbols:
Less than: <
Less than or equal to: ≤
Greater than: >
Greater than or equal to: ≥
Examples:
Q: For x  5 ,
i.
Write the solution set in set-builder notation.
A:
ii. Write the solution set in interval notation.
A:
iii. Graph the solution set.
A:
Q: For x  1,
i.
Write the solution set in set-builder notation.
A:
ii. Write the solution set in interval notation.
A:
iii. Graph the solution set.
A:
3
Q: For 3 y  2  10 ,
i.
Solve.
A:
ii. Write the solution set in set-builder notation.
A:
iii. Write the solution set in interval notation.
A:
iv.
Graph the solution set.
A:
Q: Solve 7 x  42 .
A:
Q: Solve 2x  3  17 .
A:
Q: Solve 3( x  2)  1  6 x  7 .
A:
4
Chapter 5: Polynomials
5.1 Exponents and Scientific Notation
 An exponent indicates repeated multiplication. It tells how many times the base is
used as a factor.
Example: 35  3  3  3  3  3 .
 Note: An exponent only corresponds to what is directly in front of it, unless there
are parentheses. If there are parentheses, it corresponds to everything inside the
parentheses.
Examples:
Q: Simplify: 53 .
A:
Q: Simplify: 53 .
A:
Q: Simplify:  5  .
A:
3

Powers of Quotients
*Powers of a Quotient: To raise a quotient to a power, raise the numerator and
the denominator to that power. For any numbers m and n , and any natural
number n ,
n
x
xn
   n ,
y
 y
where y  0 .
Examples:
2
4
Q: Simplify:   .
5
A:
3
2
Q: Simplify:   .
3
A:
5

Zero Exponents
*Zero exponents: Any nonzero base raised to the 0 power is 1. For any nonzero
real number x ,
x 0  1.
Examples:
Q: Simplify: 60 .
A:
Q: Simplify: 30 .
A:
Q: Simplify: 3  7  .
A:
0

Negative Integer Exponents
*Negative exponents: For any nonzero real number x and any integer n ,
x n 
1
1
and  n  x n .
n
x
x
In words, x  n is the reciprocal of x n .
Examples:
Q: Simplify: 7 3 .
A:
Q: Simplify: 6m  .
A:
2
Q: Simplify: 4 x 12 .
A:
2
2
Q: Simplify:   .
5
A:
6

Converting from Scientific to Standard Notation
 Scientific notation: A positive number is written in scientific notation when it
is written in the form N  10 n , where 1  N  10 and n is an integer.
*Large numbers (greater than 10) have positive exponents.
Small numbers (less than 1) have negative exponents.
*Converting from scientific to standard notation:
a. If the exponent is positive, move the decimal point the same number of
places to the right as the exponent.
b. If the exponent is negative, move the decimal point the same number of
places to the left as the absolute value of the exponent.
Examples:
Q: Write the number in standard notation: 4.632  10 6 .
A:
Q: Write the number in standard notation: 8.3994  10 6 .
A:

Writing Numbers in Scientific Notation
Examples:
Q: Write the number in scientific notation: 843,000,000 .
A:
Q: Write the number in scientific notation: 0.0000000349 .
A:
Q: Write the number in scientific notation: 600,340,000 .
A:
Q: Write the number in scientific notation: 0.000030540 .
A:
7
5.2 Introduction to Polynomials

Polynomials
 A polynomial is a single term or a sum of terms in which all variables have
whole-number exponents. No variable appears in the denominator.
 Polynomials are classified according to the number of terms they have. A
polynomial with exactly one term is called a monomial; exactly two terms, a
binomial; and exactly three terms, a trinomial. Polynomials with four or more
terms have no special names.
Polynomials:
Monomials
 6x
5x 2 y
29
Binomials
3u 3  4u 2
18a 2 b  4ab
Trinomials
 5t 2  4t  3
27 x 3  6 x  2
 29 z 17  1
a 2  2ab  b 2
 A term is a product or quotient of numbers and/or variables. A single number
or variable is also a term.
 The numerical factor of a term is called the coefficient of the term.
 The degree of a term of a polynomial in one variable is the value of the
exponent on the variable. If a polynomial is in more than one variable, the
degree of a term is the sum of the exponents on the variables. The degree of a
nonzero constant is 0.
 The degree of a polynomial is the same as the highest degree of any term of
the polynomial.
Examples:
Q: Describe the polynomial, 8 x7  6 xy 2  3 , using the following:
a) Is it a polynomial?
b) Is it a monomial, binomial, trinomial or does it have no special name?
c) What are the terms?
d) What are the coefficients of the terms?
e) What are the degrees of the terms?
f) What is the degree of the polynomial?
A:
8

Evaluating Polynomials
Examples:
Q: Evaluate 2 x 2  7 x  3 for x  2 .
A:
Q: Evaluate 5 x3  x 2 y  7 z for x  3 , y  4 , and z  1 .
A:
Q: The polynomial 7 r 2  2r  6 describes the voltage in a circuit, where r
represents the resistance in the circuit.
a. Find the voltage if the resistance is 6 ohms.
b. Find the voltage if the resistance is 8 ohms.
A:
9

Writing Polynomials in Descending Order
Examples:
Q: Write x5  x12  6 x3 in descending order.
A:
Q: Write 16 y  7 y 5  22  6 y 3 in descending order.
A:

Simplifying Polynomials by Combining Like Terms
Examples:
Q: Combine like terms: 0.9 x 4  0.7 x 6  0.2 x3  1.5 x 6  x3 .
A:
Q: Combine like terms: 34 x121  3x13  42 x13  x 211  4 x121  15 x13 .
A:
10
5.3 Adding and Subtracting Polynomials

Adding Polynomials
*Adding polynomials: To add polynomials, combine their like terms.
Examples:
Q: Add: 8 x 7  9 x 5  2 x 3  x  9 x 7  8x 5  x .
A:

 

 

 

Q: Add: 14 x14  40 y 5   7 x14  8xy  5 y 3  8x14  10 y 5 .
A:
Q: Write the expression for the perimeter in simplest form.
2x + 3
5x – 2
A:
11

Subtracting Polynomials
*Subtracting the polynomials: To subtract two polynomials, change the signs of
the terms of the polynomial being subtracted, drop the parentheses and combine
like terms.
Examples:
Q: Subtract: 14 x 5  3x  8   16 x 5  8x 3  9 x  2 .
A:

 

 


Q: Subtract: 8 x 2  10 x  7  9 x 2  7 x  8 .
A:
Q: The polynomial 10.55m  14.75n  27.50 p describes the revenue a pet store
generates from the sale of three different litter boxes. The expression
5.73m  8.26n  15.22 p describes the cost the store pays to sell each of the
products. Write a polynomial in simplest form that describes the store’s net
profit.
A:
12