Algebraic Expressions Preserve Structure

Algebra begins when arithmetic is organized around quantities whose values may vary or remain unknown. An expression is not merely a calculation waiting to happen; it preserves relationships that remain true across many values.

Learning objectives

You will identify terms and factors, evaluate expressions, combine like terms, distribute and factor, and determine when two expressions are equivalent.

Anatomy of an expression

In $3x^2-5x+7$, the terms are $3x^2$, $-5x$, and $7$. In $3x^2$, the coefficient is $3$ and $x^2$ is the variable factor. Addition separates terms; multiplication creates factors.

Equivalence is stronger than agreement once

Expressions are equivalent on a domain if they produce the same value for every allowed input. The distributive property gives

$$3(x+4)=3x+12.$$

Checking one input can disprove equivalence but cannot prove it. A valid algebraic property proves the identity for all inputs.

Factoring reverses distribution

From $ab+ac=a(b+c)$, the shared factor $a$ can be extracted. Thus

$$6x^2+9x=3x(2x+3).$$

Factored and expanded forms are equivalent but reveal different structure: expanded form shows terms; factored form shows zeros and multiplicative components.

Domain restrictions survive simplification

The expression $(x^2-1)/(x-1)$ simplifies to $x+1$ only when $x\ne1$. Cancellation removes a factor, not the original restriction. Equivalent formulas may have different apparent domains unless restrictions are carried forward.

Check your understanding

1. Simplify $4(2x-3)-3(x+5)$.
2. Are $(x+2)^2$ and $x^2+4$ equivalent?

Show the reasoning

1. $8x-12-3x-15=5x-27$.
2. No. $(x+2)^2=x^2+4x+4$. For example, at $x=1$ the values are $9$ and $5$.