Algebra Review

Precise definition

Algebra studies quantities and relationships using symbols under defined operations. An expression has a value; an equation asserts two expressions are equal for specified variable values; an identity is true throughout a domain. Solving means finding the complete solution set while preserving equivalence or checking any one-way transformations.

Notation and mathematical language

Key language includes coefficient, term, factor, exponent, root, domain, function, and inverse operation. The distributive law $a(b+c)=ab+ac$ and exponent laws such as $a^ma^n=a^{m+n}$ have conditions; for $a^m/a^n$, require $a\ne0$.

Conceptual picture

Algebraic fluency is structural: recognize common factors, balance equations, choose useful forms, and connect equations to graphs. Memorizing moves such as 'send it across' hides the operation applied to both sides and causes sign errors.

Conditions and key results

Cancellation applies to factors, not terms. Squaring, multiplying by a variable expression, or clearing denominators can introduce or lose information unless restrictions and checks are retained. Real square roots are nonnegative, while solving $x^2=a$ may give two roots when $a>0$.

A reliable strategy

1. State the domain and restrictions, then simplify each side without crossing an equality sign carelessly.
2. Choose factoring, expansion, substitution, graphing, or inverse operations according to structure.
3. Record transformations and solve all resulting cases.
4. Substitute candidates into the original problem and connect the answer to a graph or context.

Fully worked example

Interpretation and application

Algebra supports geometry, calculus, probability, finance, and modelling. Different forms reveal different features: factored form exposes zeros, vertex form extrema, and standard form coefficients. Equivalent forms must retain the same domain.

Common mistakes

Verification and reasonableness

• Substitute into the original expression or equation.
• Compare two forms numerically at several allowed inputs and prove equivalence algebraically.
• Use graph intercepts, signs, degree, and estimated magnitude as independent checks.

Practice

1. Factor $x^2-9$.
2. Solve $4x-7=13$.
3. State the restriction for $(x+1)/(x-4)$.

Further deduction

A high-yield review order is arithmetic and fractions, exponent laws, linear equations, factoring, rational expressions, radicals, and functions. Each stage depends on the earlier one. Use interleaved practice after initial repair so the question no longer announces which method to use; method selection is part of algebraic competence.

Polynomial division supplies a useful bridge between symbolic manipulation and function structure. For a polynomial $p$, the remainder on division by $x-a$ is $p(a)$, so $p(a)=0$ exactly when $x-a$ is a factor. For example, $p(x)=x^3-4x^2+x+6$ has $p(2)=8-16+2+6=0$, hence $x-2$ is a factor; division gives $x^2-2x-3=(x-3)(x+1)$. The three zeros are therefore $-1,2,3$. Expanding the factors reconstructs the original polynomial, while the product of roots and leading coefficient can check the constant term. This theorem turns a numerical substitution into a justified structural conclusion and also clarifies why testing one value does not prove a polynomial identity: an identity requires equality for every allowed input, whereas the factor theorem draws a conclusion about one specific root.

Related topics

• Common Algebra Errors
• Domain and Range
• Graphs of Functions