Математика25 мин

Algebraic Expressions

Collect like terms, expand brackets, factorise, and simplify algebraic fractions — the building block under almost every later math topic.

Тәжірибе

What is an algebraic expression?

An algebraic expression is a combination of numbers, variables, and operations. Unlike an equation it has no equals sign — you can simplify it, but not solve it.

Reorder in standard form (highest power first, alphabetical inside) before simplifying — pairing like terms becomes automatic.

Expanding brackets

The distributive property is the engine: $a(b+c) = ab + ac$ . Apply it twice for two binomials.

(a+b)(c+d) = ac + ad + bc + bd

FOIL — First, Outer, Inner, Last.

(a+b)^2 = a^2 + 2ab + b^2

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

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

Difference of squares.

Expand and simplify

Expand $(2x-3)(x+4)$ .

FOIL: $2x \cdot x + 2x \cdot 4 + (-3) \cdot x + (-3) \cdot 4$ .
Multiply: $2x^2 + 8x - 3x - 12$ .
Collect: $2x^2 + 5x - 12$ .

Жауап: $2x^2 + 5x - 12$

Factorising

Pull out highest common factor: $6x^2 - 9x = 3x(2x - 3)$ .
Quadratic $x^2 + bx + c$ : find two numbers that multiply to $c$ and add to $b$ .
Difference of squares: $a^2 - b^2 = (a-b)(a+b)$ .

Factor a quadratic

Factor $x^2 + 7x + 12$ .

Need two numbers with product 12 and sum 7.
Try (3, 4): $3 \cdot 4 = 12$ , $3 + 4 = 7$ . ✓
$x^2 + 7x + 12 = (x+3)(x+4)$ .

Жауап: $(x+3)(x+4)$

When the leading coefficient is not 1 (e.g. $2x^2 + 5x - 3$ ), use the AC-method or grouping. Don't guess.

Simplifying algebraic fractions

Treat algebraic fractions like numeric ones: factor numerator and denominator, then cancel common **factors** — never terms across a + or − sign.

Simplify

Simplify $\dfrac{x^2 - 9}{x^2 - x - 6}$ .

Factor numerator: $(x-3)(x+3)$ .
Factor denominator: $(x-3)(x+2)$ .
Cancel $(x-3)$ : $\dfrac{x+3}{x+2}$ .

Жауап: $\dfrac{x+3}{x+2}$ , valid when $x \neq 3, -2$

Жаттығу

Simplify $(3a - 2b)(3a + 2b)$ .

Жауап пен түсініктемені көру

Жауап: $9a^2 - 4b^2$

Difference of squares with $a' = 3a$ , $b' = 2b$ .

Осы тақырыпты жаттығу