NC Math 1: a complete guide to expressions and operations

What this strand demands

This guide covers Expressions and Operations on the NC Math 1 EOC, drawing on three NCSCOS domains: Seeing Structure in Expressions (NC.M1.A-SSE), Arithmetic with Polynomials (NC.M1.A-APR), and The Real Number System (NC.M1.N-RN). These sit inside the Number and Quantity and Algebra block, which the NCDPI specifications weight at about 36 to 40 percent of the test, the largest reporting block. The skills here, reading expressions, rewriting and factoring, operating on polynomials, and handling radicals and exponents, are the algebraic foundation for every later module. Each dot-point page has its own practice: interpreting expressions, rewriting expressions using structure, polynomial operations, factoring quadratics, radicals and rational exponents, and rational and irrational numbers.

Reading and rewriting expressions

The A-SSE standards are about structure. First, name the parts: an expression is built from terms (separated by $+$ or $-$), each made of factors (multiplied), with a coefficient (number on a variable) and exponents. In context, a coefficient is usually a rate, a constant is a starting value, and in an exponential $ab^x$ the $a$ is the initial value and $b$ is the growth or decay factor. Second, use the structure to rewrite: pull a common factor, spot a difference of squares $a^2 - b^2 = (a - b)(a + b)$, or recognize a perfect-square trinomial. Equivalent forms carry different information, and the factored form is prized because it reveals the zeros.

Polynomial operations

Polynomials are closed under addition, subtraction, and multiplication (A-APR.1), just like the integers. To add or subtract, combine like terms, and when subtracting, distribute the minus sign to every term first. To multiply, distribute every term of one polynomial across the other; for two binomials this is the four-product FOIL pattern. The result is always another polynomial, which is the closure idea.

Factoring quadratics

Factoring is the workhorse skill. Always factor out the GCF first. Then match a pattern: for $x^2 + bx + c$ find two numbers that multiply to $c$ and add to $b$; for a difference of squares write the two-binomial form; for $ax^2 + bx + c$ with $a \ne 1$ factor by grouping. The factored form reveals the zeros because $(x - r)(x - s) = 0$ gives $x = r$ or $x = s$.

Radicals and rational exponents

A rational exponent is a radical: $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m}$. This follows from keeping the exponent properties consistent: product rule $a^m a^n = a^{m+n}$, quotient rule $\frac{a^m}{a^n} = a^{m-n}$, power rule $(a^m)^n = a^{mn}$, and negative exponent $a^{-n} = \frac{1}{a^n}$. To evaluate $a^{m/n}$, take the root first, then the power: $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$.

Rational and irrational numbers

A rational number is a ratio of integers (decimal terminates or repeats); an irrational number is not (decimal never repeats). The rationals are closed under addition and multiplication. But a rational plus an irrational is irrational, and a nonzero rational times an irrational is irrational, both proved by assuming the result is rational and isolating the irrational part to reach a contradiction. Note $\sqrt{9} = 3$ is rational; a radical is irrational only when the radicand is not a perfect square.

How this strand is examined

• Gridded response. Simplify a polynomial, evaluate a rational-exponent expression, or enter a zero after factoring. Exact-match scoring.
• Multiple choice and multiple select. Identify the coefficient or growth factor, the equivalent factored form, or which numbers are irrational.
• Calculator-inactive. Much of this strand, factoring, combining terms, simplifying, fits the no-calculator section.

Check your knowledge

Work these as you would for credit on the EOC.

1. In $C = 15n + 60$, interpret the $15$ and the $60$. (2 points)
2. Factor $x^2 - 36$. (1 point)
3. Simplify $(4x^2 - 2x + 5) - (x^2 + 3x - 1)$. (2 points)
4. Multiply $(x + 7)(x - 4)$. (1 point)
5. Factor $x^2 + 9x + 20$ and state the zeros of $y = x^2 + 9x + 20$. (2 points)
6. Write $\sqrt[3]{x^2}$ using a rational exponent. (1 point)
7. Evaluate $16^{1/2}$. (1 point)
8. Is $5 + \sqrt{7}$ rational or irrational, and why? (1 point)