Precalculus Study Guide: Algebraic Expressions, Equations, and Polynomials

Algebraic Expressions and Simplification

Basic Operations with Numbers and Variables

Algebraic expressions involve numbers, variables, and arithmetic operations. Simplifying these expressions is a foundational skill in precalculus, requiring the application of arithmetic rules, properties of exponents, and order of operations.

• Addition and Subtraction: Combine like terms (terms with the same variable and exponent).

• Multiplication: Use the distributive property and properties of exponents.

• Division: Simplify fractions and reduce to lowest terms.

• Order of Operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: Simplify $2 \cdot (5 + 2^2) + 6$ First, calculate the exponent: $2^2 = 4$ Then, inside p

• arentheses: $5 + 4 = 9$ Multiply: $2 \cdot 9 = 18$ Add: $18 + 6 = 24$

Fractions and Least Common Multiple (LCM)

Operations with Fractions

When adding or subtracting fractions, it is necessary to find a common denominator, often the least common multiple (LCM) of the denominators.

• LCM: The smallest positive integer that is a multiple of two or more numbers.

• Example: The LCM of 18 and 12 is 36.

• Adding Fractions: $\frac{1}{18} + \frac{7}{12}$ Find LCM (36), convert fractions: $\frac{2}{36} + \frac{21}{36} = \frac{23}{36}$

Opposites and Reciprocals

Definitions and Examples

• Opposite: The opposite of a number $a$ is $-a$.

• Reciprocal: The reciprocal of a nonzero number $a$ is $\frac{1}{a}$.

• Example: The opposite of $19$is$-19$; the reciprocal of$\frac{5}{2}$is$\frac{2}{5}$.

Solving Linear Equations

Isolating the Variable

To solve linear equations, isolate the variable on one side using inverse operations.

• Example: $2x = 4x + 17$ Subtract $4x$ from both sides: $2x - 4x = 17$ $-2x = 17$ Divide by $-2$: $x = -\frac{17}{2}$

Exponents and Radicals

Properties and Simplification

Exponents and radicals are used to express repeated multiplication and roots. Simplifying expressions with exponents often involves applying the laws of exponents.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Radical to Exponent: $\sqrt[n]{a^m} = a^{m/n}$

• Example: $125^{2/3} = (5^3)^{2/3} = 5^2 = 25$

Polynomials: Addition, Subtraction, and Multiplication

Combining and Expanding Polynomials

Polynomials are algebraic expressions with terms consisting of variables raised to whole-number exponents. Operations include addition, subtraction, and multiplication.

• Standard Form: Write terms in descending order of exponents.

• Example: $(7y^3 - 2y^2 + 2y - 10) + (3y^3 - 9y + 4) = 10y^3 - 2y^2 - 7y - 6$

• Multiplying Binomials: $(x + 3)(x + 8) = x^2 + 11x + 24$

Factoring Polynomials

Common Methods

Factoring is the process of expressing a polynomial as a product of its factors. Common methods include factoring out the greatest common factor, difference of squares, and the AC method for trinomials.

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

• Trinomial Factoring: $x^2 + bx + c = (x + p)(x + q)$ where $p$ and $q$ are numbers such that $p + q = b$ and $pq = c$

• Example: $x^2 - 81 = (x + 9)(x - 9)$

Solving Quadratic Equations

Factoring and the Quadratic Formula

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Set each factor equal to zero and solve for $x$.

• Quadratic Formula: For $ax^2 + bx + c = 0$, the solutions are:

$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

• Example: Solve $x^2 - 8x - 2 = 0$ using the quadratic formula: $a = 1$, $b = -8$, $c = -2$ $x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 + 8}}{2} = \frac{8 \pm \sqrt{72}}{2} = \frac{8 \pm 6\sqrt{2}}{2} = 4 \pm 3\sqrt{2}$

Summary Table: Key Algebraic Properties

Property	Formula	Example
Product of Powers	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^2 = 2^5 = 32$
Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^3$
Power of a Power	$(a^m)^n = a^{mn}$	$(3^2)^3 = 3^6 = 729$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$2^{-3} = \frac{1}{8}$
Difference of Squares	$a^2 - b^2 = (a + b)(a - b)$	$x^2 - 9 = (x + 3)(x - 3)$
Quadratic Formula	$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$	$x^2 - 4x + 3 = 0$