Recover a Quotient and Retain a Domain Restriction
Polynomial division asks which factor times the divisor reconstructs the original polynomial, possibly with a remainder.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Divide x²−4 by x−2. Give the polynomial quotient. Candidate answers: (A) x+2 (B) x+4 (C) x²+2 (D) x−2.
- Independent check inputs: Divide x²−9 by x−3. Give the polynomial quotient. Candidate answers: (A) x+3 (B) x+9 (C) x²+3 (D) x−3.
Learn the idea
Polynomial functions connect symbolic factors with graph features. Division, remainder evaluation, real zeros and multiplicity reveal structure without requiring every coefficient to be expanded, but domains and repeated roots still need attention. Polynomial division asks which factor times the divisor reconstructs the original polynomial, possibly with a remainder. A zero remainder produces an exact quotient even though a corresponding rational expression may have an excluded input.
Here is the method on the supplied teaching case. Factor the numerator as (x−2)(x+2). Polynomial division gives quotient x+2 and remainder 0; the rational expression itself is undefined at x=2.
A polynomial quotient can be defined everywhere while the original fraction is not. Distinguish polynomial division from evaluation of a rational expression at its zero denominator; cancellation does not make that original input valid.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Divide x²−4 by x−2. Give the polynomial quotient. Candidate answers: (A) x+2 (B) x+4 (C) x²+2 (D) x−2. Solution: Factor the numerator as (x−2)(x+2). Polynomial division gives quotient x+2 and remainder 0; the rational expression itself is undefined at x=2. The final answer is x+2. A polynomial quotient can be defined everywhere while the original fraction is not. Distinguish polynomial division from evaluation of a rational expression at its zero denominator; cancellation does not make that original input valid.
Check your reasoning
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Helpful tips
- Polynomial division asks which factor times the divisor reconstructs the original polynomial, possibly with a remainder.
- A polynomial quotient can be defined everywhere while the original fraction is not.
See the answer and explanation
x+3
Factor the numerator as (x−3)(x+3). Polynomial division gives quotient x+3 and remainder 0; the rational expression itself is undefined at x=3.
Take it further
Changed-input task: Divide x²−81 by x−9. Give the polynomial quotient. Candidate answers: (A) x+81 (B) x+9 (C) x²+9 (D) x−9. Show the mathematical reason, not only the selected answer. Expected result: x+9. Factor the numerator as (x−9)(x+9). Polynomial division gives quotient x+9 and remainder 0; the rational expression itself is undefined at x=9.
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