Determine the quotient of a polynomial of degree one or two divided by a polynomial of degree one when the degree of the divisor does not exceed the degree of the dividend (TEKS A.10C).

What this topic is asking

TEKS A.10C asks for the quotient when a polynomial of degree one or two is divided by a polynomial of degree one. On STAAR Algebra I this is a focused skill in the Number and Algebraic Methods category. The dividends and divisors are deliberately small, so two reliable methods cover every case: factor and cancel when the numerator factors, and polynomial long division when it does not (or to confirm).

Method 1: factor and cancel

When the numerator factors and shares a factor with the denominator, division collapses to cancelling.

This is the method to reach for first on a multiple-choice item, because it is quick and the distractors usually reward a slip such as adding the constants. The cancellation is valid for $x \neq 3$, but Algebra I asks for the simplified quotient rather than the domain restriction.

Method 2: polynomial long division

When the numerator does not factor, divide the way you divide numbers, working term by term in descending order.

Handling a remainder

If the subtraction does not reach zero, the leftover is a remainder, written as a fraction over the divisor. Dividing $\frac{x^2 + 4x + 1}{x + 1}$ gives a quotient $x + 3$ with remainder $-2$, so the result is

On the equation editor, enter the answer in the form the question asks for; if it asks only for the quotient, the polynomial part $x + 3$ may suffice, but read the prompt, since some items want the remainder term included.

How STAAR examines polynomial division

• Multiple choice. A clean factor-and-cancel quotient, with distractors that combine constants or drop a sign.
• Equation editor. Enter the quotient (and remainder if requested) exactly; standard form, like terms combined.
• Connection to factoring. A zero remainder means the divisor is a factor of the dividend, which links directly to finding zeros of a quadratic.

A clarifying idea is that division and multiplication are inverses: if dividing $P(x)$ by $(x + 3)$ gives $Q(x)$ with no remainder, then $(x + 3) \cdot Q(x) = P(x)$, which is the check you should always run.

Setting up long division cleanly

Two setup habits prevent most long-division errors on the assessed items. First, write the dividend in standard form with every power present: if a term is missing, insert it with a zero coefficient. Dividing $\frac{x^2 - 9}{x - 3}$ is safest written as $x^2 + 0x - 9$, so the "bring down" step has a place to land and the linear coefficient is not skipped. Second, bracket each product before subtracting, because the subtraction applies to the whole line, and an unbracketed subtraction is where a sign quietly flips.

When the divisor is degree one, the quotient of a degree-two dividend is always degree one, and a degree-one dividend gives a constant. Knowing the degree of the answer in advance is a quick sanity check: if your quotient has the wrong degree, a step was dropped. This degree relationship is exactly why TEKS A.10C restricts the divisor so that its degree never exceeds the dividend's, keeping every result a genuine polynomial quotient with at most a simple remainder.

Try this

Q1. Simplify $\dfrac{x^2 - 9}{x - 3}$. [1 point]

• Cue. $\frac{(x - 3)(x + 3)}{x - 3} = x + 3$.

Q2. Divide $\dfrac{2x^2 + 7x + 3}{x + 3}$. [2 points]

• Cue. Factor $2x^2 + 7x + 3 = (x + 3)(2x + 1)$; cancel to get $2x + 1$.