Virginia SOL Algebra I: a complete guide to expressions and operations (A.EO)

What this category demands

This guide covers the Expressions and Operations reporting category, the A.EO standards, about 12 of the 50 operational items on the Virginia Algebra I SOL. It is the smallest blueprint block, but its skills (simplifying, exponent laws, radicals, polynomial operations, factoring, and equivalence) are the machinery the rest of the test runs on, so they repay study many times over. Each dot-point page has its own practice: properties and simplifying expressions, exponents and scientific notation, radicals and rational exponents, polynomial operations, factoring polynomials, and equivalent expressions.

Properties and the order of operations

Simplify in the order of operations (grouping, exponents, multiply/divide left to right, add/subtract left to right), and justify each move with a property of real numbers: commutative (reorder), associative (regroup), distributive ($a(b + c) = ab + ac$), identity (add $0$ or times $1$), and inverse (opposite sums to $0$, reciprocal multiplies to $1$). Simplify an algebraic expression by distributing then combining like terms, and evaluate by substituting with parentheses so signs and exponents stay correct.

Exponents and scientific notation

The laws of exponents combine like bases: add to multiply ($x^a x^b = x^{a+b}$), subtract to divide ($x^a / x^b = x^{a-b}$), multiply for a power of a power ($(x^a)^b = x^{ab}$). A zero exponent is $1$ and a negative exponent is a reciprocal ($x^{-n} = 1/x^n$, still positive in value). Scientific notation $a \times 10^n$ uses $1 \le a < 10$: positive $n$ for large numbers, negative $n$ for small.

Radicals and rational exponents

Simplify a radical by removing the largest perfect-square factor ($\sqrt{72} = 6\sqrt{2}$) and use $\sqrt{ab} = \sqrt{a}\sqrt{b}$. A rational exponent is a radical in disguise: $x^{m/n} = \sqrt[n]{x^m}$, denominator the root and numerator the power.

Polynomial operations

Add and subtract by combining like terms (distribute the minus first when subtracting). Multiply with the distributive property: monomials term by term, binomials by FOIL. Know the special products: $(a + b)(a - b) = a^2 - b^2$ and $(a \pm b)^2 = a^2 \pm 2ab + b^2$.

Factoring

Factoring is multiplying in reverse, so always check by expanding. The checklist: GCF first, then count terms. Two terms, try a difference of squares $a^2 - b^2 = (a + b)(a - b)$. Three terms, try a perfect-square trinomial, then the general method (two numbers that multiply to $c$ and add to $b$ for $x^2 + bx + c$, or to $ac$ and add to $b$ with grouping for $ax^2 + bx + c$).

Equivalent expressions

Two expressions are equivalent when they have the same value for every input. Simplify both to standard form to test it, or substitute a value as a quick check. Use the form that helps: expanded shows degree and constant term, factored shows the zeros. Rewriting with the properties never changes value, only appearance.

How this category is examined

• Fill-in-the-blank. Simplify, multiply, or factor and type the result in standard or factored form.
• Multiple choice. Identify equivalent expressions, simplified radicals, or named properties.
• Matching and drag-and-drop. Pair equivalent forms, order solution steps, or build a product term by term.

Check your knowledge

Work these as you would for credit on the online test.

1. Simplify $4(2x - 3) - (x + 5)$. (2 points)
2. Simplify $\dfrac{15x^6}{5x^2}$. (1 point)
3. Write $0.00072$ in scientific notation. (1 point)
4. Simplify $\sqrt{50}$. (1 point)
5. Write $x^{3/4}$ as a radical. (1 point)
6. Multiply $(x - 5)(x + 2)$. (2 points)
7. Expand $(2x + 3)^2$. (2 points)
8. Factor $x^2 + 3x - 28$. (2 points)
9. Factor $9x^2 - 16$. (1 point)
10. Is $2(x + 4)$ equivalent to $2x + 4$? (1 point)