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Polynomials Practice Problems

Practice evaluating polynomials at a given value of x, drilling addition, subtraction, multiplication, and monomial division along the way, with instant feedback across three difficulty levels.

  1. 1.

    If and , find .

  2. 2.

    If and , find .

  3. 3.

    If and , find .

  4. 4.

    If and , find .

  5. 5.

    If and , find .

  6. 6.

    If and , find .

  7. 7.

    If and , find .

  8. 8.

    If and , find .

  9. 9.

    If and , find .

  10. 10.

    If and , find .
Written by
MPhil in Mathematics, Founder of Mathovia
Updated

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Polynomials Practice Problems

Adding, subtracting, multiplying, and dividing polynomials are covered in full in the Polynomials lesson; this practice set is where that skill turns into speed. Every problem asks you to evaluate a polynomial expression at a given value of — you still have to combine or expand it correctly, but the final answer is always one checkable number.

How the difficulty levels work

DifficultyWhat it drillsExample
EasyEvaluate a trinomial directly at
MediumAdd or subtract two polynomials, then evaluate
HardMultiply two binomials, or divide by a monomial, then evaluate at

Easy evaluates a single trinomial at a chosen integer — direct substitution and order of operations.

Medium gives two polynomials, and , and asks for their sum or difference evaluated at a chosen — combine like terms first, or evaluate each one separately and add; both reach the same answer.

Hard alternates between evaluating a product of two binomials (FOIL, then substitute) and evaluating the result of dividing a polynomial by a monomial — every division problem is built so it comes out exact, with no remainder.

Worked examples

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

Easy. Evaluate at .

. Square the negative number before multiplying: , not .

Medium. With and , find .

and , so . Adding first, at , gives the same .

Hard. Evaluate at .

Directly: . Expanded: , and .

Using the practice problems

Choose the difficulty and the number of questions, and a new set is ready right away. Every answer here is a whole number — negative results are common, so don’t forget the sign.

  • Check Answers checks every problem at once, flagging exactly what to redo
  • Show Answers displays the complete key immediately, useful for marking offline practice
  • Print Worksheet formats the set for printing, useful for a classroom handout or offline review

The rules these practice problems drill

RuleStatement
Combining like termsAdd or subtract coefficients only; the exponent never changes
Distributing a negative sign, applied to every term when subtracting
Multiplying binomials
Dividing by a monomialDivide every term separately

See the Polynomials lesson for worked derivations of each of these, plus the special product patterns this practice set doesn’t test directly.

Common mistakes to watch for

Forgetting to distribute the negative sign on a Medium subtraction problem. means every term of flips sign before adding — not just its first term.

Adding exponents when combining like terms. , not — combining like terms only adds coefficients.

Substituting before simplifying the sign of a negative x. When , means , a positive number — a very common sign slip.

Dividing only the first term when dividing by a monomial. Every term in the numerator needs the same division applied to it.

Where to go next

Once these feel automatic, the Algebra Equation Practice Problems shifts from evaluating expressions to solving for directly. For the full reasoning and special product patterns behind every rule drilled here, see the Polynomials lesson.

Frequently Asked Questions

Why does every problem ask me to evaluate at a specific x, instead of just simplifying?+

Simplifying a polynomial produces an expression, like "x²+8x+15", which can't be automatically checked as a single number. Evaluating at a chosen x still requires you to correctly add, subtract, multiply, or divide first — it just turns the final answer into one checkable value.

For the Medium problems, do I need to combine P and Q first, or can I evaluate each one separately?+

Both give the same answer, since evaluating is a linear operation: equals either way. Combining first is usually faster; evaluating each separately is a good way to double-check your work.

How do I know the division problems come out evenly?+

Every Hard-tier division problem is built backward from a whole-number quotient, so the numerator is always an exact multiple of the monomial denominator — there's never a remainder to worry about.

What's the difference between (P+Q)(x) and P(x)+Q(x)?+

Nothing — they're the same value, just written two ways. treats the sum as a single new function evaluated at ; evaluates each one first, then adds. Both are standard function notation for the same result.

Do I need Integer Exponents before this practice set?+

Yes — every operation here leans on the Product and Quotient Rules for exponents. If combining or multiplying terms feels shaky, Integer Exponents is the right lesson to review first.

More practice problems

Integer Exponents Practice Problems Algebra Equations Practice Problems Order of Operations Practice Problems (PEMDAS)