Algebra: Quadratic Expressions and Factorization

Study Notes

A quadratic expression is a polynomial expression of degree two, meaning the highest power of the variable (usually x) is two.
The general form of a quadratic expression is: ax^2 + bx + c, where a, b, and c are constants.
Examples of quadratic expressions: x^2 + 4x + 4, 2x^2 - 3x - 1, x^2 - 2x - 3

Factorising a quadratic expression means expressing it as a product of two binomials.
The general form of a factorised quadratic expression is: (x + p)(x + q), where p and q are constants.
To factorise a quadratic expression, find two numbers whose product is the constant term (c) and whose sum is the coefficient of the linear term (b).
Examples of factorising quadratics:
- x^2 + 5x + 6 = (x + 3)(x + 2)
- x^2 - 4x - 3 = (x - 3)(x + 1)

A cubic expression is a polynomial expression of degree three, meaning the highest power of the variable (usually x) is three.
The general form of a cubic expression is: ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Examples of cubic expressions: x^3 + 2x^2 - 7x - 1, 2x^3 - 3x^2 - 5x + 2, x^3 - 2x^2 - 3x + 1
Factoring cubic expressions can be more complex than quadratic expressions, and may require the use of algebraic methods such as the Rational Root Theorem.

Quadratic expressions are polynomials of degree two, where the highest power of the variable (usually x) is two.
The general form of a quadratic expression is: ax^2 + bx + c, where a, b, and c are constants.
Examples of quadratic expressions include: x^2 + 4x + 4, 2x^2 - 3x - 1, and x^2 - 2x - 3.

Factorising a quadratic expression involves expressing it as a product of two binomials.
The general form of a factorised quadratic expression is: (x + p)(x + q), where p and q are constants.
To factorise a quadratic expression, find two numbers whose product is the constant term (c) and whose sum is the coefficient of the linear term (b).
Examples of factorising quadratics include:
- x^2 + 5x + 6 = (x + 3)(x + 2)
- x^2 - 4x - 3 = (x - 3)(x + 1)

Cubic expressions are polynomials of degree three, where the highest power of the variable (usually x) is three.
The general form of a cubic expression is: ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Examples of cubic expressions include: x^3 + 2x^2 - 7x - 1, 2x^3 - 3x^2 - 5x + 2, and x^3 - 2x^2 - 3x + 1.
Factoring cubic expressions can be more complex than quadratic expressions and may require the use of algebraic methods such as the Rational Root Theorem.