Add, subtract, and multiply polynomials, understanding that polynomials form a system closed under these operations (TN A1.A.APR.A.1).

What this topic is asking

Standard A1.A.APR.A.1 covers the three basic operations on polynomials, addition, subtraction, and multiplication, and the idea that the result is always another polynomial (the system is closed). The graded skills are combining like terms, distributing the subtraction sign correctly, and multiplying with the distributive property (including the FOIL shortcut for two binomials).

Adding and subtracting: combine like terms

Like terms have the same variable raised to the same power: $3x^2$ and $-5x^2$ are like terms, but $3x^2$ and $3x$ are not. To add or subtract polynomials, add the coefficients of like terms.

The danger is subtraction. $(A) - (B)$ means subtract every term of $B$, so the sign of each term in $B$ flips.

Multiplying: distribute everything

To multiply, apply the distributive property: each term of the first polynomial multiplies each term of the second. For two binomials, FOIL (First, Outer, Inner, Last) is the same idea in a fixed order. Multiply coefficients and add exponents on matching variables ($x \cdot x = x^2$).

Closure: why the result is still a polynomial

Adding, subtracting, or multiplying polynomials never introduces a variable in a denominator, a root, or a negative or fractional exponent, so the answer is always another polynomial. This mirrors how the integers are closed under addition, subtraction, and multiplication: stay inside the system. The exam sometimes asks this directly, for example whether $(x^2 + 1)(x - 3)$ is a polynomial (yes), and the answer follows from closure.

How TNReady examines this topic

• Equation response. Type the sum, difference, or product in standard form, scored by exact match, so a single sign slip costs the point.
• Multiple choice. Choose the product of two binomials, with sign-error and missed-middle-term distractors.
• Drag and drop. Assemble the terms of a product into the correct expression.

A clarifying idea is that multiplication and factoring are inverse operations: expanding $(x - 4)(x + 4)$ to $x^2 - 16$ is the reverse of factoring $x^2 - 16$. Fluency in one direction directly supports the other.

Standard form and degree

Always present an answer in standard form, with terms ordered from the highest power down to the constant. The degree of the polynomial is the highest exponent, and the leading coefficient is the number on that term. Writing $x^3 - x^2 - 2x + 8$ rather than $8 - 2x + x^3 - x^2$ is not just neatness: exact-match scoring may expect standard form, and the degree and leading coefficient (which control end behavior) are easiest to read off the ordered form. When a product has a missing power, for example no $x$ term, leave a gap rather than inventing a zero term unless the item asks for placeholders.

Try this

Q1. Simplify $(3x^2 - x) + (x^2 + 4x - 5)$. [1 point]

• Cue. $4x^2 + 3x - 5$.

Q2. Expand $(2x + 1)^2$. [2 points]

• Cue. $(2x + 1)(2x + 1) = 4x^2 + 4x + 1$ (a perfect-square trinomial).