1 Fundamentals of Algebra Real Numbers Polynomials

1 Fundamentals of Algebra Real Numbers Polynomials
Factoring Polynomials Rational Expressions Integral Exponents Solving Equations Rational Exponents and Radicals Quadratic Equations Inequalities and Absolute Value

1.1 Real Numbers

The Set of Real Numbers We use real numbers everyday to describe various quantities, such as temperature, salary, annual percentage rate, shoe size, grade point average, and so on. Some of the symbols we use to represent real numbers are To construct the set of real numbers, we start with the set of natural numbers: N = {1, 2, 3, …} To this set we can adjoin other numbers, such as the zero, to create the set of whole numbers: W = {0, 1, 2, 3, …} By adjoining the negatives of the natural numbers, we obtain the set of integers: I = {…, –3, –2, –1, 0, 1, 2, 3, …}

Q = {a/b | a and b are integers, b ≠ 0}
The Set of Real Numbers Next, we consider the set Q of rational numbers, numbers of the form a/b, where a and b are integers and b ≠ 0. Using set notation we write Q = {a/b | a and b are integers, b ≠ 0} Note that I is contained in Q, since each integer may be written in the form a/b, with b = 1. Thus, we say that I is a proper subset of Q, which can expressed symbolically as I  Q However, Q is not contained in I since fractions such as 1/2 and 23/25 are not integers. We can show the relationship of all these sets as follows: N  W  I  Q

The Set of Real Numbers Finally, we obtain the set of real numbers by adjoining the set of rational numbers to the set of irrational numbers (Ir). Irrational numbers are those that cannot be expressed in the form of a/b, where a, b are integers (b ≠ 0). Examples of irrational numbers are and so on. Thus, the set R = Q  Ir comprising all rational numbers and irrational numbers is called the set of real numbers.

The Set of Real Numbers Q Ir I W N
The set of all real numbers consists of the set of rational numbers plus the set of irrational numbers: Q I W N Ir

Representing Real Numbers as Decimals
Every real number can be written as a decimal. A rational number can be represented as either a repeating or terminating decimal. For example, 2/3 is represented by the repeating decimal … which may also be written , where the bar above indicates that the 6 repeats indefinitely. The number 1/2 is represented by the terminating decimal 0.5 When an irrational number is represented as a decimal, it neither terminates nor repeats. For example,

Representing Real Numbers in the Number Line
We can represent real numbers geometrically by points on a real number, or coordinate, line: Arbitrarily select a point on a straight line to represent the number 0. This point is called the origin. If the line is horizontal, then choose a point at a convenient distance to the right of the origin to represent the number 1. The distance between the 0 and the 1 determines the scale of the number line. Origin – 4 – 3 – 2 – 1 1

Representing Real Numbers in the Number Line
We can represent real numbers geometrically by points on a real number, or coordinate, line: The point representing each positive real number x lies x units to the right of 0, and the point representing each negative real number x lies – x units to the left of 0. Thus, real numbers may be represented by points on a line in such a way that corresponding to each real number there is exactly one point on the line, and vice versa. Origin Negative Direction Positive Direction – 4 – 3 – 2 – 1 1 p

Operations with Real Numbers
Two real numbers may be combined to obtain a real number. The operation of addition, written +, enables us to combine any two numbers a and b to obtain their sum, denoted a + b. Another operation, multiplication, written ·, enables us to combine any two real numbers a and b to form their product, the number a · b (more simply written ab).

Rules of Operation for Real Numbers
Properties of Addition Property Example 1. a + b = b + a = 3 + 2 2. a + (b + c) = (a + b) + c 4 + (2 + 3) = (4 + 2) + 3 3. a + 0 = a = 6 4. a + (– a) = (– 5) = 0

The operation of subtraction is defined in terms of addition. If we let – b be the additive inverse of b, the expression a + ( – b) may be written in the more familiar form a – b and we say that b is subtracted from a.

The property of associativity does not apply for subtraction. For Example: a – (b – c) ≠ (a – b) – c 4 – (2 – 3) ≠ (4 – 2) – 3 The property of commutativity does not apply for subtraction either. For Example: a – b ≠ b – a 2 – 3 ≠ 3 – 2

Properties of Negatives Property Example 1. – (– a) = a – (– 6) = 6 2. (– a)b = (– ab) = a(– b) (– 3)4 = (– 3 · 4) = 3(– 4) 3. (– a)(– b) = ab (– 3)(– 4) = 3 · 4 4. (– 1)a = – a (– 1)5 = – 5

Properties of Multiplication Property Example 1. ab = ba 2 · 3 = 3 · 2 2. a(bc) = (ab)c 4 · (2 · 3) = (4 · 2) · 3 3. a · 1 = a 5 · 1 = 5 4.

The operation of division is defined in terms of multiplication. Recall that the multiplicative inverse of a nonzero real number b is 1/b, also written as b–1. that a is divided by b. Zero does not have a multiplicative inverse since division by zero is not defined.

The property of associativity does not apply for division. For Example: a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c 4 ÷ (2 ÷ 3) ≠ (4 ÷ 2) ÷ 3 The property of commutativity does not apply for division either. For Example: a ÷ b ≠ b ÷ a 2 ÷ 3 ≠ 3 ÷ 2

Properties of Quotients Property Example

1.2 Polynomials

Exponents Expressions such as 25, (–3)2, and (1/4)4 are exponential expressions. More generally, if n is a natural number and a is a real number, then an represents the product of a and itself n times.

Exponents If a is a real number and n is a natural number, then
an = a · a · a · a · · · · · a 34 = 3 · 3 · 3 · 3 The natural number n is called the exponent, and the real number a is called the base.

Exponents Examples:

Property 1 If m and n are natural numbers is a is a real number, then
am · an = am + n 32 · 33 = = 35

Property 1 Examples:

Polynomial in One Variable
A polynomial in x is an expression of the form anxn + an-1xn a1x + a0 where n is a nonnegative integer and a0, a1, … , an are real numbers, with an ≠ 0. The expressions akxk in the sum are called the terms of the polynomial. The numbers a0, a1, a2, … , an are called the coefficients of 1, x, x2, …, xn respectively. The coefficient an of xn (the highest power in x) is called the leading coefficient of the polynomial. The nonnegative integer n gives the degree of the polynomial.

Example: Consider the polynomial: The terms of the polynomial are –2x5, 8x3, – 6x2, 3x, and 1, respectively. The coefficients of 1, x, x2, x3, and x5 are 1, 3, – 6, 8, –2, respectively. The leading coefficient of the polynomial is –2. The degree of the polynomial is 5.

A polynomial having just one term is called a monomial. For example: A polynomial having exactly two terms is called a binomial. A polynomial having exactly three terms is called a trinomial. A polynomial consisting of one constant term a0 is called a constant polynomial. For example: – 8

Polynomial in Several Variables
Most of the terminology used for a polynomial in one variable is applicable to polynomials in several variables. But the degree of a term in a polynomial in several variables is obtained by adding the powers of all variables in the term, and the degree of the polynomial is given by the highest degree of all its terms. For example, the polynomial is a polynomial in the two variables x and y. It has five terms with degrees 7, 4, 3, 1, and 0, respectively. Accordingly, the degree of the polynomial is 7.

Adding and Subtracting Polynomials
Constant terms and terms having the same variables and exponents are called like or similar terms. Like terms may be combined by adding or subtracting their numerical coefficients. For example, we can use the distributive property of the real number system to perform and

Adding and Subtracting Polynomials
Examples: Remove parentheses Group like terms together Combine the terms

Multiplying Polynomials
To find the product of two polynomials, we again use the distributive property for real numbers. For example, to compute the product we use the distributive law to obtain

Examples Find the product of Solution Distributive property Distributive property Multiply terms Combine the terms

Examples Multiply Solution Distributive property Distributive property Multiply terms Arrange terms in order of descending powers of x

Special Products Here are some commonly used products of polynomials:
Formula Example

Factoring Polynomials
1.3 Factoring Polynomials

Factoring Factoring a polynomial is a process of expressing it as a product of two or more polynomials. For example, by applying the distributive property we may write 3x2 – x = x(3x – 1) and we say that x and 3x – 1 are factors of 3x2 – x.

Common Factors The first step in factoring a polynomial is to check if it contains any common factors. If it does, the common factor of highest degree is factored out. For example, the greatest common factor of is 2a because Thus, we can factor out 2a as follows:

Some Important Factoring Formulas
Having checked for common factors, the next step in factoring a polynomial is to express the polynomial as the product of a constant and/or one or more prime polynomials. The following formulas are very useful in this and should therefore be memorized.

Some Important Factoring Formulas
Example

Examples Factor the expression Solution

Trial-and-Error Factorization
The factors of the second-degree polynomial px2 + qx + r where p, q, and r are integers, have the form where ac = p, ad + bc = q, and bd = r. Since only a limited number of choices are possible, we use a trial-and-error method to factor polynomials having this form.

Example Factor the expression x2 – 2x – 3 Solution
We first observe that, since the coefficient of x2 is 1, the only possible first-degree terms are Next, we observe that the product of the constant terms is (– 3). This gives us the following possible factors:

Coefficients of outer terms Coefficients of outer terms
Example Factor the expression x2 – 2x – 3 Solution We have two possible sets of factors: Now, the coefficient of x in the polynomial is (– 2). We multiply the coefficients of the inner terms and the outer terms and add them to see which set of factors yields (– 2): Coefficients of outer terms Outer terms Inner terms Coefficients of outer terms

Coefficients of outer terms Coefficients of outer terms
Example Factor the expression x2 – 2x – 3 Solution We have two possible sets of factors: Now, the coefficient of x in the polynomial is (– 2). We multiply the coefficients of the inner terms and the outer terms and add them to see which set of factors yields (– 2): Coefficients of outer terms Outer terms Coefficients of outer terms Inner terms

Example Factor the expression x2 – 2x – 3 Solution
We have two possible sets of factors: Now, the coefficient of x in the polynomial is (– 2). We multiply the coefficients of the inner terms and the outer terms and add them to see which set of factors yields (– 2): Thus, we conclude that the correct factorization is

Examples Use trial and error to factorize the following expressions:

Factoring by Regrouping
Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out. Examples Factor the expression Solution Rearrange the terms Factor the first two terms Factor the common term x + 1

Factoring by Regrouping
Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out. Examples Factor the expression Solution Factor 2a from the first two terms and b from the second two terms Factor the common term x + y

1.4 Rational Expressions

Rational Expressions Quotients of polynomials are called rational expressions. Examples Because division by zero is not allowed, the denominator of a rational expression must not equal zero. Thus, in the first example, x ≠ – 3/2, and in the second example, y ≠ 4x.

Simplifying Rational Expressions
A rational expression is simplified, or reduced to lowest terms, if its numerator and denominator have no common factors other than 1 and –1. If a rational expression does contain common factors, we use the properties of the real number system to write This process if often called “canceling common factors.” To indicate this process, we often write a slash through the common factors being cancelled: (a, b, and c are real numbers, and bc ≠ 0)

Factorize numerator and denominator Cancel any common factors
Examples Simplify the expression Solution Factorize numerator and denominator Cancel any common factors

Factorize numerator and denominator Cancel any common factors
Examples Simplify the expression Solution Factorize numerator and denominator Rewrite the term 1 – 2x in the form – (2x – 1) Cancel any common factors

Multiplication and Division
If P, Q, R, and S are polynomials, then Multiplication Example (Q, S ≠ 0)

Multiplication and Division
If P, Q, R, and S are polynomials, then Division Example (Q, R, S ≠ 0)

Examples Perform the indicated operation and simplify:

Examples Perform the indicated operation and simplify: 2

Addition and Subtraction
If P, Q, R, and S are polynomials, then Addition Example (R ≠ 0)

Addition and Subtraction
If P, Q, R, and S are polynomials, then Subtraction Example (R ≠ 0)

Examples Perform the indicated operation and simplify:

Compound Fractions A fractional expression that contains fractions in its numerator or denominator is called a compound fraction. The techniques used to simplify rational expressions may be used to simplify these fractions.

Examples Simplify the expression:

1.5 Integral Exponents

Exponents Recall that if a is a real number and n is a natural number, then an = a · a · a · a · · · · · a The natural number n is called the exponent, and the real number a is called the base. n factors

Examples Write each of the numbers below without using exponents:

Zero Exponent For any nonzero real number a, a0 = 1
The expression 00 is not defined. Examples:

Exponential Expressions With Negative Exponents
If a is any nonzero real number and n is a positive integer, then

Examples Write each of the numbers below without using exponents:

Properties of Exponents
If m and n are integers and a is a real number, then 1. am · an = am + n 32 · 33 = = 35 2. 3. (am)n = amn (x4)3 = x4·3 = x12 4. (ab)n = an · bn (2x)4 = 24x4 = 16x4 5.

Examples Simplify the expression, writing your answer using positive exponents only:

1.6 Solving Equations

Equations An equation is a statement that two mathematical expressions are equal. The following are examples of equations:

Equality Properties of Real Numbers
Let a, b, and c be real numbers. 1. If a = b, then a + c = b + c Addition property and a – c = b – c Subtraction property 2. If a = b, and c ≠ 0, then ac = bc Multiplication property Division property

Linear Equations A linear equation in the variable x is an equation that can be written in the form ax + b = 0 where a and b are constants with a ≠ 0. A linear equation in x is also called a first degree equation in x or an equation of degree 1 in x.

Examples Use the equality properties of real numbers to solve the equation

Rational Exponents and Radicals
1.7 Rational Exponents and Radicals

nth Root of a Real Number
If n is a natural number and a and b are real numbers such that then we say that a is the nth root of b.

For n = 2 and n = 3, the roots are commonly referred to as the square roots and the cube roots, respectively. Examples: – 2 and 2 are square roots of 4 because (– 2)2 = 4 and 22 = 4. – 3 and 3 are square roots of 9 because (– 3)2 = 9 and 32 = 9. – 4 and 4 are square roots of 16 because (– 4)2 = 16 and 42 = 16.

How many real roots does a real number b have? 1. When n is even, the real nth roots of a positive real number b must come in pairs: one positive and one negative. For example, the real fourth roots of 81 include – 3 and 3. To avoid ambiguity we define the principal nth root of a positive number when n is even to be the positive root

How many real roots does a real number b have? 2. When n is even and b is a negative real number, there are no real nth roots of b. For example, if b = – 9 and the real number a is a square root of b, then by definition a2 = – 9. But this is a contradiction since the square of a real number cannot be negative, so b has no real roots in this case.

How many real roots does a real number b have? 3. When n is odd, then there is only one real nth root of b. For example, the cube root of – 64 is – 4.

Radicals We use the notation called a radical, to denote the principal nth root of b. The symbol is called a radical sign, and the number b within the radical sign is called the radicand. The positive integer n is called the index of the radical. For square roots (n = 2), we write instead of

Examples Determine the number of roots of the real number Solution
Here b > 0, n is even, and there is one principal root. Thus,

Here b = 0, n is odd, and there is one root. Thus,

Here b < 0, n is odd, and there is one root. Thus,

Here b < 0, n is even, and so no real root exists. Thus, is not defined.

Rational Exponents If n is a natural number and b is a real number, then (If b < 0 and n is even, b1/n is not defined)

Rational Exponents If m/n is a rational number reduced to its lowest terms (m, n natural numbers), then or, equivalently, whenever it exists.

Examples Simplify the expressions:

Negative Exponents If m/n is a rational number reduced to its lowest terms (m, n natural numbers), then

Examples Simplify the expressions:

Properties of Radicals
If m and n are natural numbers and a is a real number for which the indicated roots exist, then Property Example

Simplifying Radicals An expression involving radicals is simplified if the following conditions are satisfied: The powers of all factors under the radical sign are less than the index of the radical. The index of the radical has been reduced as far as possible. No radical appears in a denominator. No fraction appears within a radical.

Examples Simplify the radical:

Rationalizing the Denominator
The process of eliminating a radical from the denominator of an algebraic expression is referred to as rationalizing the denominator. This can be done by multiplying both the numerator and the denominator by the radical that we wish to eliminate. For example:

Examples Rationalize the denominator:

1.8 Quadratic Equations

Quadratic Equations A quadratic equation in the variable x is any equation that can be written in the form where a, b, and c are constants and a ≠ 0. We refer to this form as the standard form. Examples of quadratic equations in standard form:

Solving by Factoring We solve a quadratic equation in x by finding its roots. The roots of a quadratic equation in x are the values of x that satisfy the equation. The method of solving quadratic equations by factoring relies on the following zero-product property of real numbers: Zero-Product Property of Real Numbers If a and b are real numbers and ab = 0, then a = 0 , or b = 0, or both a, b = 0.

Examples Solve by factoring. x2 – 3x + 2 = 0 Solution
Factoring the equation, we find that (x – 2)(x – 1) = 0 By the zero-product property of real numbers, we have x – 2 = or x – 1 = 0 from which we see that x = 2 or x = 1 are the roots of the equation.

Examples Solve by factoring. 2x2 – 7x = – 6 Solution
Rewriting the equation in standard form, we have 2x2 – 7x + 6 = 0 Factoring the equation, we find that (2x – 3)(x – 2) = 0 By the zero-product property of real numbers, we have 2x – 3 = or x – 2 = 0 from which we see that the roots of the equation are x = 3/2 or x = 2.

Examples Solve by factoring. 4x2 – 3x = 0 Solution
Factoring the equation, we find that x(4x – 3) = 0 By the zero-product property of real numbers, we have x = or x – 3 = 0 from which we see that the roots of the equation are x = or x = 3/4

Solving by Completing the Square
Write the equation ax2 + bx + c = 0 in the form where the coefficient of x2 is 1 and the constant term is on the right side of the equation. Example

Square half of the coefficient of x. Example

Add the number obtained in step 2 to both sides of the equation, factor, and solve for x. Example

Examples Solve by completing the square: Solution 1. First write
2. Square half of the coefficient of x, obtaining

Examples Solve by completing the square: Solution
3. Add 9/64 to both sides of the equation: Factoring, we have

Examples Solve by completing the square: Solution 1. First write
2. The coefficient of x is 0, so we can skip step 2. 3. Taking the square root in both sides, we have

The Quadratic Formula The solutions of ax2 + bx + c = 0
(a ≠ 0) are given by

Examples Use the quadratic formula to solve Solution
The equation is in standard form, with a = 2, b = 5, and c = – 12.

Examples Use the quadratic formula to solve Solution
We first rewrite the equation in standard form from which we see that a = 1, b = 3, and c = – 8. Thus, or

Inequalities and Absolute Value
1.9 Inequalities and Absolute Value

Intervals We described the system of real numbers in Section 1.1.
We will now be interested in certain subsets of real numbers called finite intervals and infinite intervals. Finite intervals can be open, closed, or half-open.

Finite Intervals Open intervals
The set of all real numbers that lie strictly between two fixed numbers a and b is called an open interval (a, b). It consists of all real numbers x that satisfy the inequalities a < x < b. It is called “open” because neither of its endpoints is included in the interval. For example, the open interval (–2, 1) includes all the real numbers between –2 and 1, but does not include the numbers –2 and 1 themselves. Graphically:

Finite Intervals Closed intervals
The set of all real numbers between two fixed numbers a and b that includes the numbers a and b is called a closed interval [a, b]. It consists of all real numbers x that satisfy the inequalities a  x  b. It is called “closed” because both of its endpoints are included in the interval. For example, the closed interval [–1, 2] includes all the real numbers between –2 and 1, including the numbers –1 and 2 themselves. Graphically:

Finite Intervals Half-open intervals
The set of all real numbers between two fixed numbers a and b that includes only one of the endpoint numbers a or b is called a half-open or half-closed interval. The half-open interval (a, b] consist of all real numbers x that satisfy the inequalities a < x  b. For example, the half-open (–3, 1/2] includes all the real numbers between –3 and 1/2, including the number 1/2 but not including the number –3. Graphically:

Finite Intervals Half-open intervals
The set of all real numbers between two fixed numbers a and b that includes only one of the endpoint numbers a or b is called a half-open or half-closed interval. The half-open interval [a, b) consist of all real numbers x that satisfy the inequalities a  x < b. For example, the half-open [–2, 1) includes all the real numbers between –2 and 1, including the number –2 but not including the number 1. Graphically:

Infinite Intervals Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a, x  a, x < a, and x  a, respectively. For example, the infinite interval (2, ∞) satisfies x > 2 and includes all the real numbers greater than 2, but does not include the number 2 itself. Graphically: (

Infinite Intervals Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a, x  a, x < a, and x  a, respectively. For example, the infinite interval [2, ∞) satisfies x  2 and includes all the real numbers greater than 2, including the number 2 itself. Graphically: [

Infinite Intervals Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a, x  a, x < a, and x  a, respectively. For example, the infinite interval (– ∞, 1) satisfies x < 1 and includes all the real numbers less than 1, but does not include the number 1 itself. Graphically: )

Infinite Intervals Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a, x  a, x < a, and x  a, respectively. For example, the infinite interval (– ∞, 1] satisfies x  1 and includes all the real numbers less than 1, including the number 1 itself. Graphically: ]

Properties of Inequalities
Let a, b, and c be any real numbers, 1. If a < b and b < c, then a < c. Example 2 < 3 and 3 < 8, so 2 < 8.

Let a, b, and c be any real numbers, 2. If a < b, then a + c < b + c. Example Consider –5 < –3 so –5 + 2 < –3 + 2 that is, –3 < –1

Let a, b, and c be any real numbers, 3. If a < b and c > 0, then ac < bc. Example Consider –5 < – and > 0 so (–5)(2) < (–3)(2) that is, –10 < –6

Let a, b, and c be any real numbers, 4. If a < b and c < 0, then ac > bc. Example Consider –5 < – and –2 < 0 so (–5)(–2) > (–3)(–2) that is, 10 > 6

Examples Solve the inequality Solution
The solution is the set of all values of x in the interval (– ∞, 3).

The solution is the set of all values of x in the interval [2, 6).

Solving Inequalities by Factoring
The method of factoring can be used to solve inequalities that involve polynomials of degree 2 or higher. This method relies on the principle that a polynomial changes sign only at a point where its value is 0. To find the values where a polynomial is equal to 0, we set the polynomial equal to 0 and then solve for x. The values obtained can then be used to help us solve the inequality.

First, set to 0 the polynomial in the inequality and factor the polynomial: Thus, the polynomial changes signs at x = 2 and at x = 3.

Next, we construct a sign diagram for the factors of the polynomial: Since x2 – 5x + 6 > 0, we require that the product of the two factors be positive, which occurs when both factors have the same sign. The diagram shows us that the two factors have the same sign when x < 2 or x > 3. Thus, the solution set is (–∞, 2)  (3, ∞). Sign of (x – 3) – – – – – – – – – – – – – – – – – – – – – – (x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – ) (

First, set to 0 the polynomial in the inequality and factor the polynomial: Thus, the polynomial changes signs at x = – 4 and at x = 2.

Next, we construct a sign diagram for the factors of the polynomial: Since x2 + 2x – 8 < 0, we require that the product of the two factors be negative, which occurs when the two factors have the different sign. The diagram shows us that the two factors have different signs when – 4 < x < 2. Thus, the solution set is (– 4, 2). Sign of (x + 4) – – – – – – – (x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – – ( )

First, rewrite the inequality so that the right side equals 0: Thus, the quotient changes signs at x = – 1 and at x = 2.

Next, we construct a sign diagram for the factors of the quotient: Since the quotient of these factors must be positive or equal to 0, we require that these two factors have the same sign. The diagram shows us that the two factors have same sign for all values in (–∞, –1]  (2, ∞). Note that x = 2 is not included, since division by 0 is not allowed. Sign of (x + 1) – – – – – – – – – – – – – – – – (x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – – ] (

Absolute Value The absolute value of a number a is denoted by |a| and is defined by |a| = Example |5| = 5 and |-5| = 5 Geometrically |a| is the distance between the origin and the point on the number line that represents the number a

End of Chapter

1 Fundamentals of Algebra Real Numbers Polynomials

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