Key Concepts:
Simplification of Expressions:
The process of combining like terms and applying arithmetic or algebraic operations to write an expression in its simplest form.Examples:
\[ 3x + 5x = 8x \quad \text{(combine like terms)} \] \[ 2(x + 4) - 3(x - 2) = 2x + 8 - 3x + 6 = -x + 14 \quad \text{(expand and simplify)} \]Expansion:
The process of removing brackets in algebraic expressions by applying the distributive property. Each term inside the bracket is multiplied by the term outside.Examples:
\[ a(b + c) = ab + ac \] \[ (x + 2)(x + 5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10 \]Factorization:
The reverse of expansion — writing an expression as a product of its factors. Common techniques include taking out common factors, using identities, or grouping terms.Examples:
\[ 6x + 9 = 3(2x + 3) \quad \text{(common factor)} \] \[ x^2 + 5x + 6 = (x + 2)(x + 3) \quad \text{(quadratic factorization)} \] \[ a^2 - b^2 = (a - b)(a + b) \quad \text{(difference of squares)} \]Factorization Formulas:
\[ a^2 - b^2 = (a-b)(a+b) \] \[ a^2 + 2ab + b^2 = (a+b)^2 \] \[ a^2 - 2ab + b^2 = (a-b)^2 \] \[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \] \[ x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]Coefficient:
The numerical factor in a term of an algebraic expression. For example, in \(5x^2\), the coefficient is 5.Term:
A single mathematical expression involving numbers, variables, or their product. For example, \(3x\), \(-7y^2\), and \(4\) are all terms of expression \(3x-7y^2+4\).Polynomial Expression:
An algebraic expression made up of terms consisting of variables raised to whole number powers and their coefficients. The general form is: \[ a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \] where \(a_i \in \mathbb{R}\) and \(n \in \mathbb{N}_0\).Rational Expression:
An expression that can be written as the ratio of two polynomials: \[ \frac{P(x)}{Q(x)}, \quad \text{where } Q(x) \neq 0 \]Irrational Expression:
An algebraic expression that involves roots (square roots, cube roots, etc.) of variables or numbers that cannot be expressed as a ratio of polynomials. \[ \text{Example: } \sqrt{x + 1}, \quad \frac{1}{\sqrt{x - 2}} \]Example 1: Expand \((x+2)(x-3)\).
Solution:
Use the distributive property: \[ (x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6. \] Therefore, the expanded expression is \(x^2 - x - 6\).Example 2: Factorize \(xy + xz + wy + wz\).
Solution:
Group terms and factorize: \[ xy + xz + wy + wz = x(y+z) + w(y+z) = (x+w)(y+z). \] Therefore, the factorized expression is \((x+w)(y+z)\).Example 3: Factorize \(x^2 - 9\).
Solution:
Recognize the expression as a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3). \] Therefore, the factorized form is \((x - 3)(x + 3)\).Example 4: Simplify and factor \((x + 3)^2 - (x - 2)^2\)
Solution:
Step 1: Expand both squares using identities: \[ (x + 3)^2 = x^2 + 6x + 9, \quad (x - 2)^2 = x^2 - 4x + 4 \] Step 2: Subtract the expressions: \[ x^2 + 6x + 9 - (x^2 - 4x + 4) \] Step 3: Simplify: \[ x^2 + 6x + 9 - x^2 + 4x - 4 = 10x + 5 \] Step 4: Factor the result: \[ 10x + 5 = 5(2x + 1) \] Therefore, the simplified and factored form is \( \boxed{5(2x + 1)} \).Example 4: Simplify \(\frac{2 - 18m^2}{1 + 3m}\).
Solution:
Step 1: Factorize the numerator: \[ 2 - 18m^2 = 2(1 - 9m^2). \] Recognize \(1 - 9m^2\) as a difference of squares: \[ 1 - 9m^2 = (1 - 3m)(1 + 3m). \] Thus: \[ 2 - 18m^2 = 2(1 - 3m)(1 + 3m). \] Step 2: Simplify the fraction: \[ \frac{2 - 18m^2}{1 + 3m} = \frac{2(1 - 3m)(1 + 3m)}{1 + 3m}. \] Cancel the common factor \((1 + 3m)\) (valid when \(1 + 3m \neq 0\)): \[ \frac{2(1 - 3m)(1 + 3m)}{1 + 3m} = 2(1 - 3m). \] Step 3: Expand if needed: \[ 2(1 - 3m) = 2 - 6m. \] Therefore, the simplified expression is \(2 - 6m\).