Read polynomial remainders in Book Isle: match the divisor sign to the evaluation point

A polynomial remainder is not an arbitrary sum of coefficients. Read an example with an explicit divisor in Book Isle, reconstruct dividend equals divisor times quotient plus remainder, and verify at the specified point. Work on paper or in external notes; no symbolic calculator in the reader is assumed, and one substitution alone does not prove every polynomial identity.

Copy degrees and missing terms

Record book, edition, example location and full dividend. Define P(x)=x^3-2*x+1 and list coefficients one, zero, minus two, one in descending degree. Zero holds the missing quadratic place; later terms must not slide forward. Fix the divisor as x-2, rather than recording only two. Return to the page to check unclear signs, exponents or brackets before calculating instead of substituting a more familiar exercise.

Identify quotient and remainder separately

For this example the candidate quotient is Q(x)=x^2+2*x+2 and the remainder is the constant five. Keep separate columns. A remainder below a degree-one divisor must be constant or zero, not an unfinished linear expression. Polynomial remainders do not inherit the same nonnegative convention as integer remainders; another divisor below yields a negative constant. Map the book’s letters explicitly so the quotient’s constant two is not mistaken for remainder five.

Multiply back to verify the identity

Expand the product manually: cubic remains, positive and negative quadratic terms cancel, linear terms are two x minus four x, and the constant is minus four. Adding five restores one. Write P(x)=(x-2)*(x^2+2*x+2)+5 and compare every degree. Multiplication checks the whole expression and catches omissions that a final number alone misses. A positive two x calls for revisiting multiplication by minus two, not altering the remainder to conceal the linear error.

For a long-division ledger, first place x squared in the quotient and subtract x cubed minus two x squared, leaving two x squared minus two x plus one. Next place two x and subtract two x squared minus four x, leaving two x plus one. Finally place two and subtract two x minus four to obtain five. Subtract the entire product each time, changing both signs where needed. This makes the missing quadratic placeholder part of the actual check, not a decorative initial zero.

Evaluate where the divisor vanishes

At x equal to two, the divisor vanishes and its entire product with the quotient disappears. Evaluate eight minus four plus one to obtain five. This comes from the linear-divisor identity, not because two is convenient. Note that it depends on the identity above: finding five alone does not establish the candidate quotient. If an answer key supplies the remainder, preserve your own working rather than label a copied answer as a completed derivation.

Recheck the sign after changing the divisor

For the control divisor x+2, substitute minus two, not positive two. P at minus two is minus eight plus four plus one, giving minus three. A negative constant is a valid polynomial remainder. The corresponding quotient is x squared minus two x plus two; multiplying back and adding minus three restores P. Mark that only the divisor changed. Replacing the dividend with another exercise destroys the intended comparison.

Discuss factors only after a zero remainder

Only zero remainder establishes divisibility by that linear polynomial. Remainder five does not identify a root. Here P at one is one minus two plus one, zero, so x minus one is a factor; that does not conflict with remainder five for x minus two. Attach each conclusion to its own evaluation point and divisor. This confirms one root and factor, not a list of all real or complex roots, and a small numerical residual is not automatically exact zero.

An additional check at zero gives one on the original side and minus two times two plus five, also one, on the reconstructed side. This helps catch an omitted final five but remains supplementary to expansion. An erroneous remainder four would give zero, contradicting P’s constant one; correct the working rather than change the problem.

Check placeholder zeros and negative cubes

If both divisors yield the same remainder, first check whether positive two was used twice, then whether minus two cubed became positive eight. For shifted synthetic-division steps, restore the four degree positions and the missing zero without moving powers. Correct the identified error while retaining the earlier record. If ebook formatting obscures the formula, retain its location and consult a clear authorised copy rather than claim automatic formula repair by the app.

Leave and return to explain both checks

Leave the page and return, deriving the evaluation point from each divisor before recomputing five and minus three with notes covered. Keep P, both divisors, candidate quotients, multiplied identities, evaluated values and open issues. Explain the sign for x plus two, valid negative remainders, and why numeric evaluation does not replace checking the quotient identity. Success is traceable signs, degrees and reasoning rather than memorising one remainder. Record all assumptions again for another problem.

Related guides and sources

Book Isle Euclidean algorithm: trace divisors, remainders and the stopping condition; Follow worked examples in Book Isle: track symbols, units and substitutions.

Checked on November 13, 2026: Apple public information lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the official App Store for published capabilities; verify specific controls in the installed version.

Method reference: OpenStax: polynomial division and the division identity. Samples, passages and controls are original exercises. Sources establish definitions and checking methods rather than new automatic analysis in the app. Manual records may be kept externally, and unknown fields should remain explicit. This article does not claim step-by-step testing on a physical device, private server or every output environment.