Algebraic Formulas

Basic Algebra Formulas

Foundational formulas for simplifying and solving algebraic expressions.

Commutative Property
\( a + b = b + a \)
\( a \times b = b \times a \)

Addition and multiplication are commutative.

Associative Property
\( (a + b) + c = a + (b + c) \)
\( (a \times b) \times c = a \times (b \times c) \)

Grouping does not affect addition or multiplication.

Distributive Property
\( a(b + c) = ab + ac \)

Distributes multiplication over addition.

Identity Property
\( a + 0 = a \)
\( a \times 1 = a \)

Zero and one are identity elements for addition and multiplication.

Inverse Property
\( a + (-a) = 0 \)
\( a \times \frac{1}{a} = 1 \ (a \neq 0) \)

Additive and multiplicative inverses yield identity elements.

Difference of Squares
\( a^2 - b^2 = (a - b)(a + b) \)

Factoring a difference of two squares.

Square of Binomial
\( (a + b)^2 = a^2 + 2ab + b^2 \)
\( (a - b)^2 = a^2 - 2ab + b^2 \)

Expansion of squared binomials.

Cube of Binomial
\( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)
\( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)

Expansion of cubed binomials.

Sum of Cubes
\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

Factoring a sum of two cubes.

Difference of Cubes
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

Factoring a difference of two cubes.

Fourth Power of Binomial
\( (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \)
\( (a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4 \)

Expansion of fourth-power binomials.

Perfect Square Trinomial
\( a^2 + b^2 + 2ab = (a + b)^2 \)
\( a^2 + b^2 - 2ab = (a - b)^2 \)

Recognizing perfect square trinomials.

Double Distributive Property
\( (a + b)(c + d) = ac + ad + bc + bd \)

Expanding product of two binomials (FOIL method).

Difference of Fourth Powers
\( a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) = (a - b)(a + b)(a^2 + b^2) \)

Factoring a difference of fourth powers.

Sum of Squares (Complex)
\( a^2 + b^2 = (a + bi)(a - bi) \)

Factoring sum of squares using complex numbers (\( i^2 = -1 \)).

General Binomial Expansion (n=5)
\( (a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 \)

Expansion of fifth-power binomial.

Zero Product Property
\( ab = 0 \implies a = 0 \text{ or } b = 0 \)

If product is zero, at least one factor is zero.

Substitution Property
If \( a = b \), then \( a \) can replace \( b \) in any expression.

Equal values can be substituted in expressions.

Quadratic Equations

Formulas for solving quadratic equations of the form \( ax^2 + bx + c = 0 \).

Quadratic Formula
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Solves for roots of a quadratic equation.

Discriminant
\( D = b^2 - 4ac \)

Determines the nature of roots (real, repeated, or complex).

Sum of Roots
\( x_1 + x_2 = -\frac{b}{a} \)

Sum of the roots of a quadratic equation.

Product of Roots
\( x_1 x_2 = \frac{c}{a} \)

Product of the roots of a quadratic equation.

Vertex of Parabola
\( x = -\frac{b}{2a} \)
\( y = f\left(-\frac{b}{2a}\right) \)

Coordinates of the vertex of the parabola.

Exponents and Powers

Rules for manipulating expressions with exponents.

Product Rule
\( a^m \times a^n = a^{m+n} \)

Multiply powers with same base by adding exponents.

Quotient Rule
\( \frac{a^m}{a^n} = a^{m-n} \)

Divide powers with same base by subtracting exponents.

Power Rule
\( (a^m)^n = a^{m \times n} \)

Raise a power to another power by multiplying exponents.

Zero Exponent
\( a^0 = 1 \ (a \neq 0) \)

Any non-zero base raised to zero is one.

Negative Exponent
\( a^{-n} = \frac{1}{a^n} \)

Negative exponent inverts the base.

Fractional Exponent
\( a^{\frac{1}{n}} = \sqrt[n]{a} \)

Fractional exponent represents a root.

Product to Power
\( (ab)^n = a^n \times b^n \)

Distribute power to each factor in a product.

Quotient to Power
\( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

Distribute power to numerator and denominator.

Arithmetic and Geometric Progressions

Formulas for sequences and series.

Arithmetic Progression (AP)

nth Term
\( a_n = a + (n-1)d \)

Finds the nth term (a = first term, d = common difference).

Sum of n Terms
\( S_n = \frac{n}{2} [2a + (n-1)d] \)
\( S_n = \frac{n}{2} (a + l) \)

Sum of first n terms (l = last term).

Geometric Progression (GP)

nth Term
\( a_n = a \times r^{n-1} \)

Finds the nth term (a = first term, r = common ratio).

Sum of n Terms (r ≠ 1)
\( S_n = \frac{a (1 - r^n)}{1 - r} \)
\( S_n = \frac{a (r^n - 1)}{r - 1} \)

Sum of first n terms for r < 1 or r > 1.

Sum of Infinite GP (|r| < 1)
\( S = \frac{a}{1 - r} \)

Sum of infinite terms when |r| < 1.

Binomial Theorem

Formulas for expanding binomial expressions.

Binomial Expansion
\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Expands a binomial raised to power n.

Binomial Coefficient
\( \binom{n}{k} = \frac{n!}{k! (n-k)!} \)

Calculates coefficients in binomial expansion.

General Term
\( T_{k+1} = \binom{n}{k} a^{n-k} b^k \)

Finds the (k+1)th term in the expansion.

Linear Equations

Formulas for solving systems of linear equations.

Slope-Intercept Form
\( y = mx + c \)

Linear equation with slope m and y-intercept c.

Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Calculates slope between two points.

Point-Slope Form
\( y - y_1 = m (x - x_1) \)

Linear equation using a point and slope.

Distance Formula
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Distance between two points in a plane.

Midpoint Formula
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Midpoint between two points.

Cramer's Rule (2x2)
For \( ax + by = e \), \( cx + dy = f \)
\( x = \frac{ed - bf}{ad - bc} \)
\( y = \frac{af - ec}{ad - bc} \)

Solves a 2x2 system of linear equations.

Polynomials

Formulas for working with polynomial expressions and their roots.

Polynomial Form
\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \)

General form of a polynomial.

Factor Theorem
If \( P(c) = 0 \), then \( (x - c) \) is a factor of \( P(x) \).

Identifies factors based on roots.

Remainder Theorem
If \( P(x) \) is divided by \( (x - c) \), the remainder is \( P(c) \).

Finds remainder of polynomial division.

Synthetic Division

Used to divide \( P(x) \) by \( (x - c) \) to find roots or simplify.

Sum of Coefficients
For \( P(x) \), sum of coefficients is \( P(1) \).

Evaluates polynomial at x = 1.

Sum of Roots (Cubic)
For \( ax^3 + bx^2 + cx + d = 0 \), sum of roots = \( -\frac{b}{a} \).

Sum of roots for a cubic polynomial.

Product of Roots (Cubic)
For \( ax^3 + bx^2 + cx + d = 0 \), product of roots = \( -\frac{d}{a} \).

Product of roots for a cubic polynomial.

Sum of Pairwise Products (Cubic)
For \( ax^3 + bx^2 + cx + d = 0 \), sum of pairwise products = \( \frac{c}{a} \).

Sum of pairwise products of roots for a cubic.

Inequalities

Key properties and formulas for solving algebraic inequalities.

Linear Inequality
\( ax + b > c \)
\( x > \frac{c - b}{a} \) (if a > 0) or \( x < \frac{c - b}{a} \) (if a < 0).

Solving linear inequalities.

Quadratic Inequality
For \( ax^2 + bx + c > 0 \), solve roots and test intervals using the quadratic formula.

Uses roots to determine solution intervals.

AM-GM Inequality
\( \frac{a + b}{2} \geq \sqrt{ab} \)

Arithmetic mean is at least geometric mean, equality when \( a = b \).

Cauchy-Schwarz Inequality
\( (a_1 b_1 + a_2 b_2)^2 \leq (a_1^2 + a_2^2)(b_1^2 + b_2^2) \)

Relates sums of products to sums of squares.

Triangle Inequality
\( |a + b| \leq |a| + |b| \)

Absolute value of sum is at most sum of absolute values.