Algebraic Expressions

Evaluating Expressions

Algebraic expressions can be evaluated by substituting given values for variables and following the order of operations.

• Key Point: Substitute the values for each variable and simplify step by step.

• Example: For where and , substitute and simplify.

Exponents and Percentages

Converting Decimals to Percentages

To convert a decimal to a percentage, multiply by 100 and add the percent symbol (%).

• Key Point: as a percentage is .

• Example: as a percentage is .

Polynomials

Writing and Simplifying Algebraic Expressions

Algebraic phrases can be translated into expressions using variables and operations.

• Key Point: "The number decreased by the sum of eighteen and this number" translates to .

• Example: "Twice a number decreased by five" is .

Classifying Polynomials

Polynomials are classified by degree and number of terms.

• Degree: The highest exponent of the variable.

• Terms: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).

• Example: is a trinomial of degree 4.

Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors, often by taking out the greatest common factor (GCF).

• Key Point: Factor out the lowest power of each variable.

• Example: .

Radical Expressions

Simplifying Radical Expressions

Radical expressions can be simplified by factoring under the radical and reducing.

• Key Point: can be rationalized by multiplying numerator and denominator by .

• Example: .

Rational Exponents

Rational exponents represent roots and powers. Negative exponents indicate reciprocals.

• Key Point: .

• Example: .

Linear Equations

Solving Word Problems with Linear Equations

Linear equations can be used to solve real-world problems involving investments, rates, and totals.

• Key Point: Set up equations based on the problem statement and solve for unknowns.

• Example: If is invested at 5% and at 3%, and total interest is , set up .

Rational Equations

Inverse Variation

Inverse variation occurs when one variable increases as the other decreases, such that their product is constant.

• Key Point: , where is speed, is diameter, and is constant.

• Example: If , solve for unknowns.

Imaginary and Complex Numbers

The Imaginary Unit

The imaginary unit is defined as . Powers of cycle every four terms.

• Key Point: , , .

• Example: .

Complex Numbers

Complex numbers are of the form , where and are real numbers.

• Key Point: Division by can be simplified using .

• Example: .

Quadratic Equations

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

• Example: For , , , .

Square Root Property

For equations of the form , solutions are .

• Example: has solutions and .

Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form .

• Key Point: Add and subtract the same value to create a perfect square trinomial.

• Example: can be written as .

Geometry in Algebra

Applying Algebra to Triangles

Algebraic equations can represent geometric properties, such as the Pythagorean theorem for right triangles.

• Key Point: For a right triangle with sides , , and hypotenuse , .

• Example: If sides are , , and , set up .

Summary Table: Polynomial Classification

Additional info:

• Some problems involve translating word problems into algebraic equations, a key skill in College Algebra.

• Imaginary and complex numbers are introduced, including operations and simplification.

• Quadratic equations are solved using multiple methods, including the quadratic formula and completing the square.