Add and subtract polynomials of degree one and degree two, and multiply polynomials of degree one and degree two, writing the result in standard form (TEKS A.10A, A.10B).

What this topic is asking

The Number and Algebraic Methods reporting category (TEKS A.10) opens with polynomial arithmetic. STAAR Algebra I expects you to add and subtract polynomials of degree one and two (A.10A) and to multiply them (A.10B), then write the result in standard form, meaning descending powers with like terms combined. These skills are the algebraic plumbing behind every quadratic on the test, so they appear both on their own (often as a quick multiple-choice or equation-editor item) and inside larger quadratic problems.

Adding and subtracting: combine like terms

Like terms have the same variable raised to the same power. You add or subtract their coefficients and keep the variable part unchanged. Addition is direct:

Subtraction is where points are lost. The subtraction sign acts on the whole second polynomial, so distribute it first:

Notice that $+x$ became $-x$ and $-7$ became $+7$. A safe routine is to rewrite subtraction as adding the opposite of every term, then combine.

Multiplying: distribute every term

Multiplication uses the distributive property. For two binomials, FOIL (First, Outer, Inner, Last) is the standard order, but a box (area) method scales to any size and prevents missed terms.

For a monomial times a polynomial, distribute once: $3x(2x^2 - x + 4) = 6x^3 - 3x^2 + 12x$. STAAR keeps degrees at one and two for the factors, so a product is at most degree two in the assessed items, with the occasional degree-three result from a monomial times a binomial.

Why standard form matters on the equation editor

The redesigned test often asks you to build the product in the equation editor rather than pick it. That item is scored by exact match against a key written in standard form, so $2x^2 + 7x - 15$ matches but $-15 + 7x + 2x^2$ or the unsimplified $2x^2 + 10x - 3x - 15$ may not. Train yourself to finish every product by ordering the powers and combining like terms, even when the question does not say "simplify".

A clarifying idea is that polynomials are closed under addition, subtraction, and multiplication: the result is always another polynomial. That is why these operations feel mechanical once the sign and like-term discipline is automatic, and it is exactly that fluency the category rewards.

Degree and leading term as a quick check

Tracking the degree of a product gives a fast sanity check. Multiplying a degree-one binomial by a degree-one binomial always produces a degree-two trinomial (unless the leading terms cancel, which they cannot for two binomials with nonzero leading coefficients), so a product of $(2x - 3)(x + 5)$ must lead with an $x^2$ term. The leading coefficient of the product is simply the product of the leading coefficients, here $2 \times 1 = 2$, and the constant term is the product of the constants, here $(-3)(5) = -15$. Checking just those two endpoints, the $x^2$ coefficient and the constant, catches a surprising share of errors before you even verify the middle term, and it is a habit worth keeping on multiselect items where several near-miss expansions are offered.

Try this

Q1. Subtract: $(7x^2 + x - 3) - (2x^2 - 4x + 5)$. [1 point]

• Cue. Distribute the minus: $7x^2 + x - 3 - 2x^2 + 4x - 5 = 5x^2 + 5x - 8$.

Q2. Multiply and write in standard form: $(x - 6)(x - 2)$. [1 point]

• Cue. $x^2 - 2x - 6x + 12 = x^2 - 8x + 12$.