Precalculus Study Notes: Review of Basic Concepts (Algebraic Expressions, Polynomials, Exponents, and Radicals)

Ch. R - Review of Basic Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables using mathematical operations. Understanding the distinction between numerical and algebraic expressions is foundational for precalculus.

• Numerical Expressions: Contain only numbers and operations. Example: $2(3) + 5$

• Algebraic Expressions: Contain variables in addition to numbers. Example: $2x + 5$

• Constant: Number without variables; its value does not change.

• Variable: Symbol representing a value that can change.

Example: Identify whether $314 + 5$ is algebraic (no variables, so it is numerical), and $2x + 5$ is algebraic (contains variable $x$).

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given values for the variables and follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: Evaluate $2x + 5$ when $x = 3$:

$2(3) + 5 = 6 + 5 = 11$

Exponents in Expressions

Exponents indicate repeated multiplication. The general form is $a^n$, where $a$ is the base and $n$ is the exponent.

• Example: $4^3 = 4 \times 4 \times 4 = 64$

• Exponent of a Power: Number of times the base is multiplied.

Simplifying Algebraic Expressions

Simplifying involves combining like terms and using the distributive property to write expressions in their simplest form.

• Terms: Parts of an expression separated by $+$ or $-$ signs.

• Like Terms: Terms with the same variable and exponent.

• Distributive Property: $a(b + c) = ab + ac$

• Example: Simplify $2x + 3x + 10 - 7x$:

$2x + 3x + 10 - 7x = (2x + 3x - 7x) + 10 = (-2x) + 10$

Polynomials

Introduction to Polynomials

Polynomials are algebraic expressions where variables have whole number exponents and are not in denominators or under radicals.

• Monomial: One term (e.g., $5x^2$)

• Binomial: Two terms (e.g., $x + 2$)

• Trinomial: Three terms (e.g., $x^2 + 3x + 2$)

• Standard Form: Terms written in descending order of exponents.

• Degree: Highest exponent of the variable in the polynomial.

• Leading Coefficient: Coefficient of the term with the highest degree.

Example: $3x^2 + 5x + 4$ is a trinomial of degree 2, leading coefficient is 3.

Adding and Subtracting Polynomials

Combine like terms to add or subtract polynomials.

• Example: $(5x^2 + 2x + 3) + (2x^2 + 7x + 8) = (5x^2 + 2x^2) + (2x + 7x) + (3 + 8) = 7x^2 + 9x + 11$

Multiplying Polynomials (FOIL Method)

Use the FOIL method for multiplying two binomials: First, Outer, Inner, Last.

• Example: $(x + 2)(x + 3)$

$= x \cdot x + x \cdot 3 + 2 \cdot x + 2 \cdot 3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Special Products: Square and Cube Formulas

Special product formulas simplify multiplication of certain polynomials.

• Square of a Binomial: $(a + b)^2 = a^2 + 2ab + b^2$

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• Cube of a Binomial: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Grouping: Group terms to factor polynomials that do not have a common factor for all terms.

• Special Product Formulas: Use formulas for difference of squares, perfect square trinomials, and sum/difference of cubes.

• AC Method: For trinomials $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.

Example: Factor $x^2 - 9 = (x + 3)(x - 3)$

Rules of Exponents

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers.

Name	Example	Rule	Description
Product Rule	$a^m \cdot a^n$	$a^{m+n}$	Add exponents when multiplying same base
Quotient Rule	$a^m / a^n$	$a^{m-n}$	Subtract exponents when dividing same base
Power Rule	$(a^m)^n$	$a^{mn}$	Multiply exponents when raising a power to a power
Zero Exponent	$a^0$	$1$	Any nonzero base to the zero power is 1
Negative Exponent	$a^{-n}$	$1/a^n$	Negative exponent means reciprocal

Simplifying Expressions with Exponents

Apply exponent rules to simplify complex expressions.

• Example: Simplify $(-5a^3b^2)/(c^{-2})$:

$= -5a^3b^2c^2$

Simplifying Radical Expressions

Simplifying Radicals by Factoring

Radicals can be simplified by factoring out perfect squares (or cubes, etc.) from under the radical sign.

• Example: $\sqrt{18x^2} = \sqrt{9x^2 \cdot 2} = 3x\sqrt{2}$

Simplifying Radicals with Variables

Apply the same principles to expressions with variables.

• Example: $\sqrt{x^4} = x^2$

Simplifying Radicals with Fractions

Radicals can be split over fractions: $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$

• Example: $\sqrt{8/2} = \sqrt{8}/\sqrt{2} = 2\sqrt{2}$

Adding & Subtracting Radicals

Combine only like radicals (same index and radicand).

• Example: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

Rationalizing Denominators

Radicals should not remain in the denominator. Multiply numerator and denominator by a suitable radical to eliminate the radical from the denominator.

• Example: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Rational Exponents

Radical expressions can be rewritten using rational exponents: $\sqrt[n]{a} = a^{1/n}$

• Example: $\sqrt{5} = 5^{1/2}$

Square Roots and Even/Odd Roots

Square roots of positive numbers have two roots (principal and negative). Even roots of negative numbers are not real; odd roots of negative numbers are negative.

• Example: $\sqrt{9} = 3$ or $-3$

• Example: $\sqrt[3]{-8} = -2$

Summary Table: Exponent Rules

Name	Example	Rule	Description
Product Rule	$a^m \cdot a^n$	$a^{m+n}$	Add exponents
Quotient Rule	$a^m / a^n$	$a^{m-n}$	Subtract exponents
Power Rule	$(a^m)^n$	$a^{mn}$	Multiply exponents
Zero Exponent	$a^0$	$1$	Any nonzero base
Negative Exponent	$a^{-n}$	$1/a^n$	Reciprocal

Summary Table: Special Product Formulas

Formula Name	Formula
Square of a Binomial	$(a + b)^2 = a^2 + 2ab + b^2$
Difference of Squares	$a^2 - b^2 = (a + b)(a - b)$
Cube of a Binomial	$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

Summary Table: Perfect Powers

Number	Square	Cube
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1000

Additional info: These notes cover foundational algebraic concepts essential for success in precalculus, including manipulation of expressions, polynomials, exponents, and radicals. Mastery of these topics is critical for understanding more advanced topics in subsequent chapters.