# Basic Concepts of Algebra

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Basic Concepts of Algebra
Chapter R

R.1 The Real-Number System
Identify various kinds of real numbers. Use interval notation to write a set of numbers. Identify the properties of real numbers. Find the absolute value of a real number.

Rational Numbers Numbers that can be expressed in the form p/q, where p and q are integers and q  0. Decimal notation for rational numbers either terminates (ends) or repeats. Examples: a) b) c) 9 d)

Irrational Numbers The real numbers that are not rational are irrational numbers. Decimal notation for irrational numbers neither terminates nor repeats. Examples: a) … b)

Interval Notation

Examples Write interval notation for each set and graph the set.
a) {x|5 < x < 2} Solution: {x|5 < x < 2} = (5, 2) ( ) b) {x|4 < x  3} Solution: {x|4 < x  3} = (4, 3] ( ]

Properties of the Real Numbers
Commutative property a + b = b + a and ab = ba Associative property a + (b + c) = (a + b) + c and a(bc) = (ab)c Additive identity property a + 0 = 0 + a = a Additive inverse property a + a = a + (a) = 0

More Properties Multiplicative identity property a • 1 = 1 • a = a
Multiplicative inverse property Distributive property a(b + c) = ab + ac

Examples State the property being illustrated in each sentence.
a) 7(6) = 6(7) Commutative b) 3d + 3c = 3(d + c) Distributive c) (3 + y) + x = 3 + (y + x) Associative

Absolute Value The absolute value of a number a, denoted |a|, is its distance from 0 on the number line. Example: Simplify. |6| = 6 |19| = 19

Distance Between Two Points on the Number Line
For any real numbers a and b, the distance between a and b is |a  b|, or equivalently |b  a|. Example: Find the distance between 4 and 3. Solution: The distance is |4  3| = |7| = 7, or |3 (4)| = |3 + 4| = |7| = 7.

R.2 Integer Exponents, Scientific Notation, and Order of Operations
Simplify expressions with integer exponents. Solve problems using scientific notation. Use the rules for order of operations.

Integers as Exponents When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product. For any positive integer n, where a is the base and n is the exponent. Example: 84 = 8 • 8 • 8 • 8 For any nonzero real number a and any integer m, a0 = 1 and Example: a) 80 = b)

Properties of Exponents
Product rule Quotient rule Power rule (am)n = amn Raising a product to a power (ab)m = ambm Raising a quotient to a power

Examples – Simplify. a) r 2 • r 5 r (2 + 5) = r 3 b)
c) (p6)4 = p -24 or 1 p24 d) (3a3)4 = 34(a3)4 = 81a12 or 81 a12 e)

Scientific Notation Use scientific notation to name very large and very small positive numbers and to perform computations. Scientific notation for a number is an expression of the type N  10m, where 1  N < 10, N is in decimal notation, and m is an integer.

Examples Convert to scientific notation. a) 17,432,000 = 1.7432  107
b) = 2.4  1010 Convert to decimal notation. a)  106 = 3,481,000 b)  105 =

Another Example Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long tunnel was completed in Construction costs were \$210 million. Find the average cost per mile.

Rules for Order of Operations
Do all calculations within grouping symbols before operations outside. When nested grouping symbols are present, work from the inside out. Evaluate all exponential expressions. Do all multiplications and divisions in order from left to right. Do all additions and subtractions in order from left to right.

Examples a) 4(9  6)3  18 = 4 (3)3  18 = 4(27)  18 = 108  18 = 90
b)

R.3 Addition, Subtraction, and Multiplication of Polynomials
Identify the terms, coefficients, and degree of a polynomial. Add, subtract, and multiply polynomials.

Polynomials Polynomials are a type of algebraic expression.
Examples: y  6t

Polynomials in One Variable
A polynomial in one variable is any expression of the type where n is a nonnegative integer, an,…, a0 are real numbers, called coefficients, and an  0. The parts of the polynomial separated by plus signs are called terms. The degree of the polynomial is n, the leading coefficient is an, and the constant term is a0. The polynomial is said to be written in descending order, because the exponents decrease from left to right.

Examples Identify the terms. 4x7  3x5 + 2x2  9
The terms are: 4x7, 3x5, 2x2, and 9. Find the degree. a) 7x5  3 5 b) x2 + 3x + 4x3 3 c)

If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms. Like Terms Unlike Terms 3y2 + 7y2 8c + 5b 4x3  2x3 9w  3y We add or subtract polynomials by combining like terms.

Examples Add: (4x4 + 3x2  x) + (3x4  5x2 + 7)
Subtract: 8x3y2  5xy  (4x3y2 + 2xy) 8x3y2  5xy  4x3y2  2xy 4x3y2  7xy

Multiplication To multiply two polynomials, we multiply each term of one by each term of the other and add the products. Example: (3x3y  5x2y + 5y)(4y  6x2y) 3x3y(4y  6x2y)  5x2y(4y  6x2y) + 5y(4y  6x2y) 12x3y2  18x5y2  20x2y2 + 30x4y2 + 20y2  30x2y2 18x5y2 + 30x4y2 + 12x3y2  50x2y2 + 20y2

More Examples Multiply: (5x  1)(2x + 5) = 10x2 + 25x  2x  5
Special Products of Binomials Multiply: (6x  1)2 = (6x)2 + 2• 6x • 1 + (1)2 = 36x2  12x + 1

R.4 Factoring Factor polynomials by removing a common factor.
Factor polynomials by grouping. Factor trinomials of the type x2 + bx + c. Factor trinomials of the type ax2 + bx + c, a  1, using the FOIL method and the grouping method. Factor special products of polynomials.

Terms with Common Factors
When factoring, we should always look first to factor out a factor that is common to all the terms. Example: x  6x2 = 6 • • 2x  6 • x2 = 6(3 + 2x  x2)

Factoring by Grouping In some polynomials, pairs of terms have a common binomial factor that can be removed in the process called factoring by grouping. Example: x3 + 5x2  10x  50 = (x3 + 5x2) + (10x  50) = x2(x + 5)  10(x + 5) = (x2  10)(x + 5)

Trinomials of the Type x2 + bx + c
Factor: x2 + 9x + 14. Solution: 1. Look for a common factor. 2. Find the factors of 14, whose sum is 9. Pairs of Factors Sum 1, 2, The numbers we need. 3. The factorization is (x + 2)(x + 7).

Another Example Factor: 2y2  20y + 48.
1. First, we look for a common factor. 2(y2  10y + 24) 2. Look for two numbers whose product is 24 and whose sum is 10. Pairs Sum Pairs Sum 1, 24 25 2, 12 14 3, 8 11 4, 6 10 3. Complete the factorization: 2(y  4)(y  6).

Trinomials of the Type ax2 + bx + c, a  1
Method 1: Using FOIL 1. Factor out the largest common factor. 2. Find two First terms whose product is ax2. 3. Find two Last terms whose product is c. 4. Repeat steps (2) and (3) until a combination is found for which the sum of the Outside and Inside products is bx.

Example Factor: 8x2 + 10x + 3. (8x + )(x + )
(8x + 1)(x + 3) middle terms are wrong 24x + x = 25x (4x + )(2x + ) (4x + 1)(2x + 3) middle terms are wrong 12x + 2x = 14x (4x + 3)(2x + 1) Correct! 4x + 6x = 10x

Grouping Method 1. Factor out the largest common factor.
2. Multiply the leading coefficient a and the constant c. 3. Try to factor the product ac so that the sum of the factors is b. 4. Split the middle term. That is, write it as a sum using the factors found in step (3). 5. Factor by grouping.

Example Factor: 12a3  4a2  16a. 1. Factor out the largest common factor, 4a. 4a(3a2  a  4) 2. Multiply a and c: (3)(4) = 12. 3. Try to factor 12 so that the sum of the factors is the coefficient of the middle term, 1. (3)(4) = 12 and 3 + (4) = 1 4. Split the middle term using the numbers found in (3). 3a2 + 3a  4a  4 5. Factor by grouping. 3a2 + 3a  4a  4 = (3a2 + 3a) + (4a  4) = 3a(a + 1)  4(a + 1) = (3a  4)(a + 1) Be sure to include the common factor to get the complete factorization. 4a(3a  4)(a + 1)

Special Factorizations
Difference of Squares A2  B2 = (A + B)(A  B) Example: x2  25 = (x + 5)(x  5) Squares of Binomials A2 + 2AB + B2 = (A + B)2 A2  2AB + B2 = (A  B)2 Example: x2 + 12x + 36 = (x + 6)2

More Factorizations Sum or Difference of Cubes
A3 + B3 = (A + B)(A2  AB + B2) A3  B3 = (A  B)(A2 + AB + B2) Example: 8y = (2y)3 + (5)3 = (2y + 5)(4y2  10y + 25)

R.5 Rational Expressions
Determine the domain of a rational expression. Simplify rational expressions. Multiply, divide, add, and subtract rational expressions. Simplify complex rational expressions.

Domain of Rational Expressions
The domain of an algebraic expression is the set of all real numbers for which the expression is defined. Example: Find the domain of Solution: To determine the domain, we factor the denominator. x2 + 3x  4 = (x + 4)(x  1) and set each equal to zero. x + 4 = 0 x  1 = 0 x =  x = 1 The domain is the set of all real numbers except 4 and 1.

Simplifying, Multiplying, and Dividing Rational Expressions
Solution: Simplify: Solution:

Another Example Multiply: Solution:

When rational expressions have the same denominator, we can add or subtract the numerators and retain the common denominator. If the denominators are different, we must find equivalent rational expressions that have a common denominator. To find the least common denominator of rational expressions, factor each denominator and form the product that uses each factor the greatest number of times it occurs in any factorization.

Example Add: Solution: The LCD is (3x + 4)(x  1)(x  2).

Complex Rational Expressions
A complex rational expression has rational expressions in its numerator or its denominator or both. To simplify a complex rational expression: Method 1. Find the LCD of all the denominators within the complex rational expression. Then multiply by 1 using the LCD as the numerator and the denominator of the expression for 1. Method 2. First add or subtract, if necessary, to get a single rational expression in the numerator and in the denominator. Then divide by multiplying by the reciprocal of the denominator.

Example: Method 1 Simplify: The LCD of the four expressions is x2y2.

Example: Method 2 Simplify:

R.6 Radical Notation and Rational Exponents
Simplify radical expressions. Rationalize denominators or numerators in rational expressions. Convert between exponential and radical notation. Simplify expressions with rational exponents.

Notation A number c is said to be a square root of a if c2 = a.
nth Root A number c is said to be an nth root of a if cn = a. The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index.

Examples Simplify each of the following: a) = 7, because 72 = 49.
b) = 7, because 72 = 49 and c) because d) because (4)3 = 64. e) is not a real number.

Let a and b be any real numbers or expression for which the given roots exist. For any natural numbers m and n (n  1): 1. If n is even, 2. If n is odd, 3. 4. 5.

Examples a) b) c) d) e) f) g) h)

Another Example Perform the operation.

The Pythagorean Theorem
The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse: a2 + b2 = c2. b c a

Example Juanita paddled her canoe across a river 525 feet wide. A strong current carried her canoe 810 feet downstream as she paddled. Find the distance Juanita actually paddled, to the nearest foot. Solution: 525 ft 810 ft x

Rationalizing Denominators or Numerators
Rationalizing the denominator (or numerator) is done by multiplying by 1 in such a way as to obtain a perfect nth power. Example: Rationalize the denominator. Example: Rationalize the numerator.

Rational Exponents For any real number a and any natural numbers m and n for which exists,

Examples Convert to radical notation and, if possible, simplify. a) b)

More Examples Convert each to exponential notation. a) b) Simplify.

R.7 The Basics of Equation Solving

A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a  0. A quadratic equation is an equation that is equivalent to one of the form ax2 + bx + c = 0, where a, b, and c are real numbers and a  0.

Equation Solving Principles
For any real numbers a, b, and c, The Addition Principle: If a = b is true, then a + c = b + c is true. The Multiplication Principle: If a = b is true, then ac = bc is true. The Principle of Zero Products: If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0. The Principle of Square Roots: If x2 = k, then or

Example: Solve 2x + 5 = 7  4(x  2)

Check: We check the result in the original equation.

Example: Solve x2  3x = 10 Write the equation with 0 on one side.

Check: x2  3x = 10 For x = 2 For x = 5

Example: Solve 5x2  25 = 0 We will use the principle of square roots.