01/14 1. Algebraic Foundations and Factoring

The Real-Number System and Its Properties

Numbers in algebra belong to a hierarchy of sets. The real-number system contains every number that can be placed on a number line.

• Natural numbers are $1,2,3,\ldots$.

• Whole numbers are $0,1,2,3,\ldots$.

• Integers include positive and negative counting numbers and zero.

• Rational numbers can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne0$.

• Irrational numbers cannot be written as a ratio of integers; examples include $\sqrt{2}$ and $\pi$.

Every integer is rational because, for example, $-6=\frac{-6}{1}$. Rational and irrational numbers are both real, but no number is both rational and irrational.

Absolute value measures distance from zero on the number line:

Therefore, $|x|\ge 0$ for every real number $x$.

The properties of real numbers explain why many algebraic rearrangements are valid. The commutative property changes order, the associative property changes grouping, identity properties preserve a value, inverse properties produce zero or one, and the distributive property connects multiplication with addition:

The distributive property can also be reversed to factor an expression, as in $3x+12=3(x+4)$.

Takeaway: Classifying numbers and recognizing the properties that govern them provides the foundation for simplifying and transforming algebraic expressions.

Expressions, Terms, and Evaluation

An algebraic expression combines numbers, variables, operation symbols, and grouping symbols, but it does not state that two quantities are equal. For example, $7x-3$ and $2a^2+5a-1$ are expressions, whereas $7x-3=11$ is an equation.

Important parts of an expression include:

• A variable represents a number whose value may vary.

• A constant is a fixed number.

• A coefficient is a numerical factor multiplying a variable.

• A term is a number, variable, or product separated from other terms by addition or subtraction.

In $-8x^2$, the coefficient is $-8$, the variable is $x$, and the exponent is $2$. In $4x^2-3x+9$, the terms are $4x^2$, $-3x$, and $9$.

Only Like terms can be combined. For example,

To evaluate an expression, substitute the assigned value for each variable and then simplify. If $x=-2$,

Parentheses are essential when substituting a negative number because $-3^2=-(3^2)=-9$, while $(-3)^2=9$.

Takeaway: Identify each part of an expression, combine only like terms, and use parentheses when substituting signed values.

Order of Operations

Use the order of operations to simplify consistently:

1. Simplify expressions inside grouping symbols.

2. Evaluate exponents and roots.

3. Perform multiplication and division from left to right.

4. Perform addition and subtraction from left to right.

Multiplication does not automatically come before division, and addition does not automatically come before subtraction; operations at the same level are evaluated from left to right.

For example,

becomes

The grouping symbols are simplified first, followed by the exponent, multiplication, and subtraction.

Takeaway: Work from the inside out, respect exponentiation, and process operations of equal priority from left to right.

Equations and Solving by Equality

An equation states that two expressions have the same value. Solving an equation means finding values of the variable that make the statement true.

A property of equality permits the same valid operation on both sides. If $a=b$, then adding or subtracting $c$ preserves equality:

Multiplication also preserves equality, and division is valid when the divisor is nonzero.

To solve $4x-7=21$, first add $7$ to both sides:

Then divide both sides by $4$:

Check the result in the original equation:

An expression can be simplified, evaluated, or factored. An equation can also be solved because it asserts a relationship between two quantities.

Takeaway: Isolate the variable by applying inverse operations to both sides, then verify the result in the original equation.

Factoring Methods and Patterns

Factoring rewrites a sum or difference as a product and reverses the distributive property. Use this general sequence:

1. Factor out the greatest common factor (GCF) first.

2. Count the terms in the remaining polynomial.

3. Select an appropriate pattern.

4. Factor completely.

5. Multiply the factors to check the result.

Common factor and grouping methods

For $12x^3+18x^2$, the GCF is $6x^2$:

For four terms, group terms with a common binomial:

Trinomials

For $x^2+bx+c$, find two numbers with product $c$ and sum $b$:

For $ax^2+bx+c$, find two numbers with product $ac$ and sum $b$, split the middle term, and factor by grouping:

Special products

A difference of squares follows the pattern

Thus,

Perfect-square trinomials follow

Cubes follow these patterns:

For example,

Takeaway: Remove the GCF before looking for patterns, match the pattern to the number of terms, and verify by multiplication.

Factoring to Solve Equations

Factoring becomes especially useful when solving polynomial equations. The zero-product principle says that if a product equals zero, at least one factor must equal zero:

Consider

Factor the left side:

Set each factor equal to zero:

Therefore,

The principle applies only when the equation is written as a product equal to zero. If the equation is not yet in that form, first move all terms to one side, factor completely, and then set each factor equal to zero.

Takeaway: Put a polynomial equation in factored zero form, solve each factor, and check every solution in the original equation.