Final Exam Review Guide: Key Concepts in Beginning Algebra

Review of Real Numbers

Rational and Irrational Numbers

Understanding the classification of numbers is fundamental in algebra. Real numbers can be divided into two main categories: rational numbers and irrational numbers.

• Rational Numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In other words, any number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples include $\sqrt{2}$, $\pi$, and $e$.

Example: $\frac{3}{4}$ is rational, while $\sqrt{5}$ is irrational.

Exponents and Polynomials

Subtracting Polynomials

Subtracting polynomials involves combining like terms and applying the distributive property. To subtract one polynomial from another, change the sign of each term in the polynomial being subtracted, then add the resulting polynomials.

• Step 1: Write both polynomials in standard form (descending order of exponents).

• Step 2: Distribute the negative sign to each term of the polynomial being subtracted.

• Step 3: Combine like terms.

Example: Subtract $(3x^2 + 2x - 5)$ from $(5x^2 - x + 4)$:

$\begin{align*} & (5x^2 - x + 4) - (3x^2 + 2x - 5) \\ &= 5x^2 - x + 4 - 3x^2 - 2x + 5 \\ &= (5x^2 - 3x^2) + (-x - 2x) + (4 + 5) \\ &= 2x^2 - 3x + 9 \end{align*}$

Exam Preparation Tips

• Review all assigned problems, focusing especially on those highlighted by your instructor.

• Practice distinguishing between rational and irrational numbers.

• Ensure you are comfortable with operations on polynomials, especially subtraction.

• Prepare your calculator and scratch paper as allowed by exam rules.

Additional info: The referenced review problems likely cover a range of topics from the beginning algebra curriculum, including equations, inequalities, graphing, and polynomials. Students should ensure they are familiar with all foundational concepts listed in the course outline.