Polynomial Long Division Practice Test

Divide polynomials, track missing powers, interpret remainders, and verify results with division identities.

Polynomial Long Division Practice Test

20 polynomial long-division questions with quotient, remainder, and verification steps.

Instant feedback · Worked explanations

Long division descent board

Repeat the cycle until the remainder is small enough.

Polynomial long division is a structured descent. At each level, divide the leading terms, multiply the divisor by the new quotient term, subtract in aligned columns, and bring down what remains. The process stops only when the remainder degree is smaller than the divisor degree.

Anchor 01Put dividend and divisor in descending powers.

Anchor 02Insert zero coefficients for every missing power.

Anchor 03Subtract aligned powers carefully at each cycle.

Anchor 04Stop only when remainder degree is below divisor degree.

1. Long division is a four-move loop

Each quotient term is created by comparing the current leading term of the dividend with the leading term of the divisor. That quotient term generates a product row, which is subtracted before the next cycle begins.

DivideDivide current leading term by divisor leading term.

MultiplyMultiply the entire divisor by the new quotient term.

SubtractSubtract the product row from aligned powers.

Bring downCarry the next unused dividend term into the new working row.

P (x) = D (x) Q (x) + R (x)

2. Missing powers must occupy real columns

Long division is column-sensitive. If a power is absent, inserting a zero coefficient preserves the correct position of every lower term.

Power alignment ledger

Every power gets a column, even when its coefficient is zero.

row

x^{4}

x^{3}

x^{2}

x

dividend

1

0

- 5

2

next row

0

1

- 5

2

Descending order firstReorder terms before beginning the division.

Zero is a placeholderA missing cubic term, for example, still needs a cubic column.

Do not shift lower powersMissing terms never move neighboring coefficients left or right.

Use the same columns in subtractionEvery product row must line up under matching powers.

x^{4} + 0 x^{3} - 5 x^{2} + 2 x + 7

3. One complete cycle: divide, multiply, subtract, bring down

The arithmetic changes from problem to problem, but the control sequence never changes.

Divide leading terms

\frac{x^{3}}{x} = x^{2}

Multiply backIf the divisor is

x - 2

, then

x^{2} (x - 2) = x^{3} - 2 x^{2}

Subtract the product rowChange every sign in the product row during subtraction, then combine aligned powers.

Bring down the next termThe new working polynomial becomes the dividend for the next cycle.

4. Degree-two divisors use the same process

A quadratic divisor does not change the algorithm. The only difference is that each quotient term is found by dividing by a quadratic leading term, and the stopping rule compares the remainder degree with two.

Leading quotient term

$\frac{6}{x^{5}} = 3 x^{3}$

That predicted term is placed in the quotient before multiplying the entire quadratic divisor back.

Stopping condition

For a quadratic divisor, a linear or constant remainder is small enough to stop, because its degree is less than $2$ .

\deg (R) < \deg (D)

5. The remainder decides when the algorithm ends

A nonzero remainder does not mean the division failed. It simply means the divisor does not divide the dividend exactly. The essential requirement is that the remainder degree be smaller than the divisor degree.

\deg (R) < \deg (D)

Exact division

R (x) = 0

The quotient reconstructs the dividend without any remainder.

Linear divisorThe remainder must be a constant or zero.

Quadratic divisorThe remainder may be linear, constant, or zero.

6. Reconstruct the dividend to verify the answer

The division algorithm gives a built-in check. Multiply the divisor by the quotient and add the remainder. The result must reproduce the original dividend term for term.

P (x) = D (x) Q (x) + R (x)

Divisor × quotient

→

Add remainder

→

Original dividend

7. The Remainder Theorem is a fast verification for linear divisors

If the divisor is $x - c$ , the remainder equals the polynomial value at $c$ . This can verify the remainder obtained by long division without repeating the entire process.

Fast remainder check

f (c) = R

Evaluate the original dividend at the divisor root.

Factor check

If $f (c) = 0$ , then $x - c$ is a factor, so long division should produce zero remainder.

8. Predict quotient degree before doing the full division

The quotient degree is determined by the difference between the dividend degree and divisor degree, assuming the dividend degree is at least as large.

Linear divisor

Drop by one

A degree- $6$ dividend divided by a degree- $1$ divisor produces a degree- $5$ quotient.

Quadratic divisor

Drop by two

A degree- $7$ dividend divided by a degree- $2$ divisor produces a degree- $5$ quotient.

Leading term

Divide leaders

The first quotient term should match the quotient of the leading terms before any lower-power work begins.

Quotient degree check

Use the degree difference

$\deg (Q) = \deg (P) - \deg (D)$ provides a structural check before the first subtraction row.

Remainder form

Show a nonzero remainder explicitly

$\frac{P (x)}{D (x)} = Q (x) + \frac{R (x)}{D (x)}$ is a standard final-answer form when the remainder is not zero.

9. Error analysis: where long division goes off track

Most long-division errors are alignment or subtraction errors rather than failures to understand the overall algorithm.

Powers misaligned

Terms must stay in fixed power columns throughout the entire division.

Missing zero placeholder

An absent power still needs a zero coefficient so later terms do not shift columns.

Only part of the divisor multiplied

Each new quotient term multiplies the entire divisor, not only its leading term.

Subtraction sign lost

Every term in the product row changes sign when that row is subtracted.

Stopped too early

If remainder degree is still at least divisor degree, another quotient term is required.

Remainder omitted

A nonzero remainder must be reported or incorporated into the final division identity.

10. Worked mini-set: structural checks before arithmetic

These examples are illustrative teaching examples, not questions copied from the test.

Example A

Missing cubic term

$x^{4} + 0 x^{3} - 3 x^{2} + 5$ keeps the cubic column visible.

Example B

First quotient term

$\frac{8}{x^{5}} = 4 x^{4}$ predicts the first quotient term.

Example C

Exact division

If reconstruction produces the dividend with $R (x) = 0$ , the divisor is an exact factor.

Example D

Nonzero remainder

A remainder is valid if its degree is already less than the divisor degree.

Example E

Remainder verification

For divisor $x - 3$ , evaluate the dividend at $x = 3$ .

Example F

Quadratic divisor

A constant or linear remainder is acceptable after division by a quadratic polynomial.

Final long-division checklist

Before accepting a quotient, inspect the power columns, subtraction rows, stopping condition, and reconstruction identity.

Are both polynomials in descending powers?Reorder them before beginning.

Did I insert zero coefficients for missing powers?This protects the alignment of every later subtraction.

Did I divide leading terms correctly?Each quotient term begins with a leading-term division.

Did I multiply back by the entire divisor?Every term in the divisor contributes to the product row.

Is the remainder degree smaller than the divisor degree?If not, the division is not finished.

Does divisor × quotient + remainder reproduce the dividend?This is the final verification.