How do you add, subtract, and multiply polynomials, and why do they behave like the integers?
Understand that polynomials are closed under addition, subtraction, and multiplication, and perform these operations (NC.M1.A-APR.1).
An NC Math 1 EOC answer on polynomial operations (NC.M1.A-APR.1): combining like terms to add and subtract, distributing the minus sign, multiplying with the distributive property and FOIL, and why polynomials form a closed system like the integers.
Reviewed by: AI editorial process; not yet individually human-reviewed
Have a quick question? Jump to the Q&A page
Jump to a section
What this topic is asking
NC.M1.A-APR.1 has two ideas. The first is conceptual: polynomials form a system analogous to the integers, meaning they are closed under addition, subtraction, and multiplication (adding, subtracting, or multiplying polynomials always gives another polynomial). The second is procedural: actually carry out those operations. On the EOC you mostly do the operations, but you may also be asked to recognize the closure idea.
Like terms and combining
Adding and subtracting polynomials is really just combining like terms.
You can only combine like terms. cannot be simplified further because the terms are unlike.
Subtraction: distribute the minus sign
Subtraction is where most points are lost, because the minus sign applies to the whole second polynomial.
Treating subtraction as "add the opposite of each term" makes the sign change automatic and prevents the classic error of only negating the first term.
Multiplication: distribute everything
To multiply polynomials, multiply each term of one by each term of the other.
- Monomial times polynomial: distribute once. .
- Binomial times binomial: four products (often remembered as FOIL: First, Outer, Inner, Last). .
FOIL is just the distributive property applied twice; for larger products, the safest method is to distribute systematically so no pair is missed.
Why closure matters
Saying polynomials are "closed" and "analogous to the integers" is not decoration. The integers are closed under addition, subtraction, and multiplication but not division ( is not an integer). Polynomials behave the same way: a sum, difference, or product of polynomials is always a polynomial, but a quotient may not be. This is the structural reason NC Math 1 does add, subtract, and multiply polynomials but leaves general polynomial division for later courses. Recognizing the analogy helps you predict which operations stay inside the system.
How the NC Math 1 EOC examines this topic
- Gridded response. Simplify a sum, difference, or product and enter the result.
- Multiple choice. Choose the simplified polynomial, often with a distractor that forgot a sign in subtraction.
- Technology-enhanced. Match expressions to their simplified forms, or select the true statement about closure.
Multiplying binomials connects directly to rewriting expressions and factoring: expanding to and factoring back to are inverse processes. Practicing both directions strengthens each.
A worked multiplication that becomes a model
Polynomial multiplication often appears in area problems, where one dimension is a binomial.
The result is a polynomial, illustrating closure: a product of two polynomials is again a polynomial.
Try this
Q1. Simplify . [1 point]
- Cue. (combine and ).
Q2. Multiply . [2 points]
- Cue. .
Exam-style practice questions
Practice questions written in the style of NCDPI exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
NC Math 1 EOC (style)2 marksSubtract: .Show worked answer →
The result is .
Distribute the minus sign across the second polynomial: . Combine like terms: terms ; terms ; constants . The most common slip is forgetting to change the sign of every term in the subtracted polynomial, especially the becoming .
NC Math 1 EOC (style)1 marksMultiply: .Show worked answer →
The product is .
Use the distributive property (FOIL): , , , . Combine the middle terms: . So . Closure means this product is again a polynomial, which A-APR.1 highlights.
Related dot points
- Use the structure of an expression to identify ways to rewrite it, and write an equivalent factored form of a quadratic to reveal zeros (NC.M1.A-SSE.2, A-SSE.3).
An NC Math 1 EOC answer on rewriting expressions using structure (NC.M1.A-SSE.2, A-SSE.3): spotting common factors, difference of squares, and perfect-square trinomials, and writing factored form to reveal the zeros of a quadratic.
- Factor quadratic expressions including common factor, trinomials, and difference of squares, and use the factored form to reveal zeros (NC.M1.A-SSE.3).
An NC Math 1 EOC answer on factoring quadratics (NC.M1.A-SSE.3): pulling out a GCF, factoring trinomials with leading coefficient 1 and greater, the difference of squares, and reading zeros from the factored form.
- Interpret the parts of a linear, exponential, or quadratic expression (terms, factors, coefficients, exponents) and interpret a multi-part expression as a combination of entities (NC.M1.A-SSE.1a, A-SSE.1b).
An NC Math 1 EOC answer on interpreting expressions (NC.M1.A-SSE.1): naming terms, factors, coefficients, and exponents, and reading what each part means in a real context for linear, quadratic, and exponential models.
- Solve quadratic equations by inspection, square roots, factoring, and the quadratic formula, writing exact solutions (NC.M1.A-REI.4a).
An NC Math 1 EOC answer on solving quadratic equations (NC.M1.A-REI.4a): the zero-product property after factoring, taking square roots, the quadratic formula, and choosing the most efficient method.
- Explain how rational exponents extend the integer-exponent properties and rewrite expressions with radicals and rational exponents (NC.M1.N-RN.1, N-RN.2).
An NC Math 1 EOC answer on radicals and rational exponents (NC.M1.N-RN.1, N-RN.2): converting between radical and exponent form, the exponent properties, and simplifying numerical and algebraic expressions.
Sources & how we know this
- North Carolina Standard Course of Study for Mathematics — NC Department of Public Instruction (2024)
- EOC NC Math 1 and NC Math 3 Test Specifications — NC Department of Public Instruction (2024)