U2 Introduction to Algebra: Essential Formulas

This section covers the fundamental concepts and formulas for algebraic manipulation, including operations with polynomials, factorization, and algebraic fractions.

1 Basic Algebraic Operations

Addition and Subtraction of Polynomials

Combine like terms. For example, $(3x^2 + 2x - 5) + (x^2 - 4x + 1)$ simplifies to $4x^2 - 2x - 4$.

$$ (a + b) + (c + d) = (a + c) + (b + d) $$
Visual Representation ofCombining Like Terms3x²x²4x²

2 Expansion and Factorization

Distributive Law (Expansion)

The process of removing brackets by multiplying each term inside the bracket by the term outside. For example, $a(b + c) = ab + ac$.

$$ a(b + c) = ab + ac $$

Difference of Two Squares

A fundamental identity used for both expansion and factorization. Note that $a$ and $b$ can be any algebraic expression.

$$ a^2 - b^2 = (a - b)(a + b) $$

Perfect Square Expansion

Crucial for solving quadratic equations and completing the square. Remember the middle term is $2ab$.

$$ (a \pm b)^2 = a^2 \pm 2ab + b^2 $$

3 Algebraic Fractions

Multiplication and Division

To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal. Always state the condition where the denominator $b, d, c \neq 0$.

$$ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \quad \text{and} \quad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} $$

Addition and Subtraction

Fractions must have a common denominator before adding or subtracting. The common denominator is typically the LCM of $b$ and $d$.

$$ \frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd} $$

4 Laws of Indices

Fundamental Rules

These rules apply when $a \neq 0$ and $m, n$ are integers (or rational numbers for roots).

$$ a^m \times a^n = a^{m+n} \quad \frac{a^m}{a^n} = a^{m-n} \quad (a^m)^n = a^{mn} $$
$$ a^{-n} = \frac{1}{a^n} \quad a^{1/n} = \sqrt[n]{a} \quad a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m} $$

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now