Subtracting Polynomials Practice Problems

The sign-distribution step gets a full explanation in the Subtracting Polynomials lesson. Here, it’s tested directly: evaluate the difference of two polynomials at a chosen value, and at Hard, sometimes in reverse order.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Subtract two binomials, evaluated at	, , find
Medium	Subtract two trinomials, evaluated at	Two degree-2 polynomials subtracted
Hard	Subtract two trinomials, sometimes in reverse order	as well as

Easy subtracts two simple binomials, keeping the arithmetic light while the sign-distribution step is still new.

Medium uses two full trinomials, so every term’s sign has to be tracked correctly across three matching-degree pairs.

Hard occasionally reverses which polynomial is subtracted from which, testing directly that and are opposites, not equal.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. , . Find .

Subtract every term of : . At , that’s . Check: .

Medium. , . Find .

, so . The became : the minus sign reaches every term.

Hard. , . Find .

Order matters: , and at that’s . Quick check: .

Using the practice problems

Pick a difficulty and however many questions you want, and a new set shows up instantly. Every answer here is a whole number, positive or negative — enter it exactly as computed.

Check Answers grades the set right away and flags exactly what needs a second attempt
Show Answers uncovers the full key right away, useful for grading paper copies
Print Worksheet prepares a neat, printable worksheet, handy for classroom or take-home use

The rules these practice problems drill

Rule	Statement
Subtraction as adding the opposite
Function subtraction
Order matters	and are always opposites, never equal

See the Subtracting Polynomials lesson for the full horizontal and vertical technique, plus why every sign has to flip.

Common mistakes to watch for

Flipping the sign of only the first term. Every term inside the second polynomial’s parentheses changes sign, not just the one next to the minus sign.

Forgetting the negative sign is multiplication by −1. It distributes across the entire polynomial being subtracted, the same way any other multiplication would.

Assuming subtraction is commutative. and are opposites of each other — always subtract in the exact order the problem states.

Combining terms before finishing the sign flip. Distribute the negative sign completely first; combining prematurely risks locking in an unflipped sign.

Where to go next

Once distributing the negative sign feels automatic, the Multiplying Polynomials Practice Problems is the natural next step, since expanding a product depends on tracking signs just as carefully. For the full technique behind every problem here, see the Subtracting Polynomials lesson.

Frequently Asked Questions

Why does every problem ask to evaluate at a specific x, instead of writing out the difference?+

Subtracting two polynomials symbolically produces another polynomial, which can't be automatically checked as a single number. Evaluating (P−Q) at a chosen x still requires correctly distributing the negative sign across every term of Q — the answer is just one checkable value instead of a full expression.

Why does Hard sometimes ask for (Q−P) instead of (P−Q)?+

Subtraction is not commutative — (P−Q) and (Q−P) are always opposites of each other, never equal — so alternating the order directly tests whether that fact is understood, not just whether the arithmetic is correct.

Is (P−Q)(x) the same as evaluating P(x) minus Q(x)?+

Yes, always — (P−Q)(x) = P(x) − Q(x) is a basic property of function subtraction, true for every polynomial and every x. That's exactly how this practice set checks each answer.

What's the most common mistake this practice set catches?+

Flipping the sign of only the first term of the polynomial being subtracted, instead of every term. Every single term inside those parentheses changes sign, not just the one written right after the minus.

How is this different from the Polynomial Practice Problems?+

The Polynomial practice set rotates between addition, subtraction, multiplication, and division. This practice set is subtraction only, at every difficulty, including the reverse-order case the broader set doesn't test.

Is there a limit on how many worksheets I can print?+

Not at all — generate freely and endlessly, at no cost, with no account required.

P−QSubtracting Polynomials Practice Problems