Basic Algebra Concepts

Like Terms and Polynomial Arithmetic

• Like Terms: Terms with the same algebraic part (e.g., the same variable and exponent). For instance, 5x and 4x are like terms because both have the variable x with the same exponent (implicitly 1). Likewise, 4x^2 and 9x^2 are like terms due to their common x^2 part.
• To combine like terms, you add or subtract their coefficients. For example, 5x + 4x becomes 9x.

Adding and Subtracting Polynomials

• When dealing with polynomials, you can only add or subtract like terms.
• Example: In 3x + 4y + 5x + 8y, you can combine 3x and 5x to get 8x, and 4y and 8y to get 12y, resulting in 8x + 12y.

Multiplying Polynomials

• To multiply polynomials, distribute each term of the first polynomial over every term of the second polynomial.
• For example, multiplying a monomial by a binomial: 7x(3x + 4) expands to 21x^2 + 28x.

Special Cases

• Squaring a Binomial: (2x - 3)^2 expands using the FOIL method, resulting in 4x^2 - 12x + 9.
• Multiplying Binomials: The FOIL method is used. Example: (3x - 4)(2x + 7) yields 6x^2 + 13x - 28.

Dividing Polynomials

• When dividing polynomials, subtract the exponents. For instance, x^9 / x^4 simplifies to x^(9-4), or x^5.

Negative Exponents and Division

• Negative exponents indicate division. For instance, x^(-3) is equivalent to 1/x^3.
• When dividing terms with exponents, subtract the exponents. If this results in a negative exponent, it indicates division by the corresponding positive exponent term.

Properties and Operation Rules

Adding Exponents

• When multiplying terms with the same base, add the exponents. Example: x^3 * x^4 becomes x^(3+4), or x^7.

Subtracting Exponents

• When dividing terms with the same base, subtract the exponents. Example: x^9 / x^4 simplifies to x^5.

Multiplying Exponents

• When raising a power to another power, multiply the exponents. For example, (x^7)^6 becomes x^(7*6), or x^42.

Solving Equations and Inequalities

• Basic techniques include isolating the variable on one side of the equation or inequality.
• Example: For x + 4 = 9, subtracting 4 from both sides yields x = 5.

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept. To graph, start at the y-intercept (b) and use the slope (m) to determine the direction and steepness of the line.