B.E.S.T. Algebra 1 EOC: a complete guide to number sense and expressions

What this strand demands

This guide covers the foundational expression skills of the B.E.S.T. Algebra 1 EOC: the laws of exponents and rational exponents (MA.912.NSO.1, in the Statistics and the Number System category) and polynomial operations, factoring, equivalent expressions, and sequences (MA.912.AR.1 and MA.912.AR.5, in the Algebra and Modeling category). These are not the highest-weighted topics on their own, but they are the machinery behind quadratics, exponentials, and modeling, so weakness here costs points across the whole test. Each dot-point page has its own practice: exponents, radicals, and rational exponents, polynomial operations, factoring polynomials, equivalent expressions, and arithmetic and geometric sequences.

The laws of exponents and rational exponents

The exponent laws are $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $(ab)^n = a^n b^n$, $a^0 = 1$, and $a^{-n} = \frac{1}{a^n}$. A rational exponent is a root: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, where the denominator is the index and the numerator is the power. To evaluate $27^{\frac{2}{3}}$, root first then raise: $\left(\sqrt[3]{27}\right)^2 = 9$. A negative exponent is a reciprocal, never a negative number.

Polynomial operations

Add and subtract by combining like terms, distributing the minus sign to every term when subtracting. Multiply with the distributive property; for two binomials, every term of one multiplies every term of the other, and the middle term is the sum of the two cross-products. Polynomials are closed under addition, subtraction, and multiplication: the result is always another polynomial. Write answers in standard form, descending powers.

Factoring

Always pull out the GCF first. Factor a trinomial $x^2 + bx + c$ by finding two numbers that multiply to $c$ and add to $b$. When $a \neq 1$, use the AC method: two numbers multiplying to $ac$, adding to $b$, split the middle term, group. Know the special patterns: difference of squares $a^2 - b^2 = (a + b)(a - b)$ and perfect-square trinomials $a^2 \pm 2ab + b^2 = (a \pm b)^2$. A sum of squares does not factor over the reals.

Equivalent expressions and interpretation

Two expressions are equivalent if they agree for every input. Produce equivalent forms by distributing, combining like terms, and factoring. In a context, the constant term is a fixed amount, a coefficient is a rate per unit, and a factor is one quantity in a product. Choosing a form is strategic: factored form shows zeros, expanded form shows the $y$-intercept.

Arithmetic and geometric sequences

An arithmetic sequence adds a common difference $d$: $a_n = a_1 + (n - 1)d$ (a linear pattern). A geometric sequence multiplies by a common ratio $r$: $a_n = a_1 \cdot r^{\,n-1}$ (an exponential pattern). Both are on the reference sheet, both use an exponent or multiplier of $n - 1$. The recursive forms, $a_n = a_{n-1} + d$ and $a_n = a_{n-1} \cdot r$, each need a stated first term.

How this strand is examined

• Equation editor. Simplify expressions, factor polynomials, or write a sequence formula, typing the exact result.
• Multiple choice and multiselect. Identify equivalent expressions, complete factorizations, or a specific sequence term, with sign and off-by-one distractors.
• Editing task choice and matching. Choose the correct exponent or pair a sequence with linear or exponential.

Check your knowledge

Work these as you would for credit on the computer-based test.

1. Simplify $\frac{x^6}{x^{-3}}$. (1 point)
2. Evaluate $32^{\frac{2}{5}}$. (1 point)
3. Simplify $(2x^2 - x + 5) - (x^2 + 3x - 2)$. (2 points)
4. Expand $(x - 6)(x + 2)$. (1 point)
5. Factor $3x^2 - 12$ completely. (2 points)
6. Factor $2x^2 + 5x - 3$. (2 points)
7. Which is equivalent to $5(x - 2) + 3$? (1 point)
8. Find $a_9$ for the arithmetic sequence with $a_1 = 4$, $d = 5$. (1 point)
9. Write the explicit formula for the geometric sequence $4, 8, 16, \dots$. (2 points)