STAAR Algebra I: a complete guide to the Number and Algebraic Methods reporting category

What this category demands

The Number and Algebraic Methods reporting category (TEKS A.10, A.11, A.12) is the smallest on the STAAR Algebra I test, about 10 percent of the points, but it is the toolbox the rest of the test reaches into. Polynomial arithmetic and factoring feed the quadratic category; the exponent laws feed the exponential category; sequences and interest connect both to growth models. This guide ties together the dot-point pages, each with its own practice: polynomial operations, factoring polynomials, dividing polynomials, exponents, radicals, and rational exponents, arithmetic and geometric sequences, and simple and compound interest.

Polynomial operations

Add and subtract by combining like terms, distributing any subtraction sign across every term of the second polynomial. Multiply with the distributive property: FOIL for two binomials, a box for larger products. Always finish in standard form, descending powers with like terms combined, because the equation editor scores an exact match. Polynomials are closed under these operations, so the result is always another polynomial.

Factoring and division

Factoring follows a fixed order: GCF first, then difference of two squares $a^2 - b^2 = (a - b)(a + b)$ or a perfect-square trinomial $a^2 \pm 2ab + b^2 = (a \pm b)^2$ (both on the reference sheet), then trinomial factoring. For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$; for $ax^2 + bx + c$, use the AC method.

Division pairs with factoring. When the numerator factors and shares a factor with the denominator, factor and cancel. When it does not, use long division, writing any leftover as a remainder over the divisor. A zero remainder means the divisor is a factor.

Exponents and radicals

The reference sheet gives the exponent laws: $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^{-n} = \frac{1}{a^n}$, and $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. A negative exponent means reciprocal, so $x^{-3} = \frac{1}{x^3}$; in a quotient, move the factor across the bar. A rational exponent is a root, so $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 4$. To simplify a square-root radical, pull out the largest perfect square: $\sqrt{72} = 6\sqrt{2}$, the same step that finishes a quadratic-formula answer.

Sequences and interest

An arithmetic sequence adds a common difference $d$: $a_n = a_1 + (n - 1)d$ (linear). A geometric sequence multiplies by a common ratio $r$: $a_n = a_1 \cdot r^{n-1}$ (exponential). A recursive rule builds each term from the previous one; an explicit rule gives any term directly. None of these formulas are on the reference sheet.

Simple interest is $I = Prt$, computed on the original principal only (linear). Compound interest (annual) is $A = P(1 + r)^t$, computed on the growing balance (exponential). Convert percents to decimals, and read whether the prompt wants the interest or the final balance.

How this category is examined

• Multiple choice and multiselect. Simplify a product or quotient, choose a complete factorization, or select all equivalent expressions. The GCF and "stop too early" distractors are standard.
• Equation editor and number entry. Build a product, a simplest-radical-form value, an $n$th-term formula, or a compound-interest balance. Exact-match scoring rewards precise, simplified work.
• Drag and drop and inline choice. Place factors into binomial templates, or classify a sequence as arithmetic or geometric.

Check your knowledge

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

1. Subtract $3x^2 - x + 5$ from $7x^2 + 2x - 1$. (1 point)
2. Multiply and write in standard form: $(2x - 5)(x + 4)$. (1 point)
3. Factor completely: $5x^2 - 45$. (1 point)
4. Simplify $\dfrac{20x^7}{4x^3}$. (1 point)
5. Write $\sqrt{98}$ in simplest radical form. (1 point)
6. An arithmetic sequence has $a_1 = 8$ and $d = 5$. Find $a_7$. (1 point)
7. Write the $n$th term of $3, 6, 12, 24, \dots$. (2 points)
8. Find the balance on $1,200 compounded annually at 5% for 2 years, to the nearest dollar. (2 points)