Find a Remainder by Evaluating the Polynomial
Dividing a polynomial by x−a leaves a constant remainder equal to its value at a.
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: Find the remainder when P(x)=x²+2x+3 is divided by x−2. Candidate answers: (A) 10 (B) 11 (C) 12 (D) 22.
- Independent check inputs: Find the remainder when P(x)=x²+2x+3 is divided by x−3. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36.
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. Dividing a polynomial by x−a leaves a constant remainder equal to its value at a. The substitution sign follows the zero of the divisor, not the minus sign visually attached to a.
Here is the method on the supplied teaching case. The remainder theorem gives P(2)=2²+2(2)+3=11. The divisor x−2 sets the substitution value to positive 2.
Identify the divisor’s zero before evaluating. The remainder is not the quotient or a root count; when it is zero, x−a is a factor, but a nonzero remainder gives a different conclusion.
- 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: Find the remainder when P(x)=x²+2x+3 is divided by x−2. Candidate answers: (A) 10 (B) 11 (C) 12 (D) 22. Solution: The remainder theorem gives P(2)=2²+2(2)+3=11. The divisor x−2 sets the substitution value to positive 2. The final answer is 11. Identify the divisor’s zero before evaluating. The remainder is not the quotient or a root count; when it is zero, x−a is a factor, but a nonzero remainder gives a different conclusion.
Check your reasoning
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Helpful tips
- Dividing a polynomial by x−a leaves a constant remainder equal to its value at a.
- Identify the divisor’s zero before evaluating.
See the answer and explanation
18
The remainder theorem gives P(3)=3²+2(3)+3=18. The divisor x−3 sets the substitution value to positive 3.
Take it further
Changed-input task: Find the remainder when P(x)=x²+2x+3 is divided by x−9. Candidate answers: (A) 101 (B) 102 (C) 103 (D) 204. Show the mathematical reason, not only the selected answer. Expected result: 102. The remainder theorem gives P(9)=9²+2(9)+3=102. The divisor x−9 sets the substitution value to positive 9.
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