CHAPTER R: Review of Algebraic Foundations

R.4 Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is essential for manipulating algebraic expressions and solving equations.

• Definition: A polynomial is an expression of the form , where the coefficients are real numbers and is a non-negative integer.

• Degree: The highest exponent of the variable in the polynomial.

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

• Example: is a polynomial of degree 4.

R.5 Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This skill is crucial for simplifying expressions and solving polynomial equations.

• Common Factoring Methods:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials

  • Difference of squares:

  • Sum/difference of cubes

• Example:

R.7 Rational Expressions

Rational expressions are fractions where the numerator and denominator are both polynomials. Simplifying and operating with rational expressions is foundational for advanced algebraic manipulation.

• Definition: A rational expression is an expression of the form , where and are polynomials and .

• Key Skills: Simplifying, multiplying, dividing, adding, and subtracting rational expressions.

• Example: (for )

R.8 Roots & Radicals

Roots and radicals involve expressions with fractional exponents or radical signs. Mastery of these concepts is necessary for solving equations and simplifying expressions.

• Definition: The n-th root of is a number such that .

• Radical Notation:

• Properties: ,

• Example:

CHAPTER 1: Equations & Inequalities

1.1 Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the first power. They are foundational for algebraic problem-solving.

• General Form:

• Solution: , where

• Example:

1.2 Quadratic Equations

Quadratic equations involve the variable raised to the second power. They are solved using factoring, completing the square, or the quadratic formula.

• Standard Form:

• Quadratic Formula:

• Example:

1.3 Complex Numbers

Complex numbers extend the real numbers to include solutions to equations like . They are written in the form .

• Definition:

• Form: , where and are real numbers

• Example:

1.4 Radical Equations

Radical equations are equations in which the variable appears under a radical sign. Solving these often involves isolating the radical and raising both sides to a power.

• Example:

• Check for extraneous solutions after solving.

1.5 Solving Inequalities

Inequalities compare two expressions and use symbols such as , , , or . Solutions are often represented on a number line or in interval notation.

• Linear Inequality Example:

• Quadratic Inequality Example: or

1.6 Equations & Inequalities Involving Absolute Value

Absolute value equations and inequalities involve expressions like . The absolute value of a number is its distance from zero on the number line.

• Definition: if , if

• Equation Example: or

• Inequality Example:

1.7 Problem Solving: Interest & Mixture

These problems apply algebraic equations to real-world contexts such as finance and chemistry.

• Simple Interest Formula: , where is interest, is principal, is rate, and is time.

• Mixture Problems: Use equations to represent the total amount and concentration of solutions mixed together.

• Example: Mixing two solutions of different concentrations to achieve a desired concentration.