Polynomial division and the remainder theorem
Find a polynomial remainder by evaluation, connect a zero remainder to a factor, and determine an unknown coefficient.
What you will tackle
- Use P(a) for division by x − a
- Connect factors, roots and remainders
- Recover an unknown coefficient from a remainder condition
Why evaluation gives the remainder
Dividing polynomial P(x) by x − a gives P(x) = (x − a)Q(x) + r, with constant remainder r. Substituting x = a makes the product zero, leaving P(a) = r. A zero remainder means x − a is a factor and a is a root.
Use the divisor's zero
The divisor x + 2 has zero −2, so evaluate P(−2), not P(2). This shortcut applies to a linear divisor; a higher-degree divisor can have a nonconstant remainder. A parameter in P can be found by equating P(a) to the specified remainder.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Find the remainder when P(x) = x³ − 2x + 5 is divided by x − 2. Is x − 2 a factor?
Show a starting hint
Start with the divisor's zero.
Compare with the explained answer
- The divisor vanishes at x = 2.
- P(2) = 8 − 4 + 5 = 9.
- Because the remainder is 9 rather than zero, x − 2 is not a factor.
Answer / response criteria: Remainder 9; not a factor.
Independent practice
Independent 1
Find the remainder of 2x³ + x² − 5x + 1 on division by x + 2. Find k if x² + kx + 6 is divisible by x − 2.
Compare with the explained answer
- Use x = −2: −16 + 4 + 10 + 1 = −1.
- Divisibility requires P(2) = 4 + 2k + 6 = 0.
- 2k = −10, so k = −5; the polynomial becomes (x − 2)(x − 3).
Answer / response criteria: Remainder −1; k = −5.
Application practice
Application 1
A proposed model P(x) = x³ + kx + 4 has remainder 10 when divided by x − 2. Find k, then check whether x + 1 is a factor and explain what that would mean for the model's output.
Compare with the explained answer
- P(2) = 8 + 2k + 4 = 10 gives k = −1.
- P(−1) = −1 + 1 + 4 = 4, so x + 1 is not a factor.
- A factor x + 1 would require output zero at input −1; the actual output is 4.
Answer / response criteria: k = −1; x + 1 is not a factor; P(−1) = 4.
What can you explain now?
A proposed model P(x) = x³ + kx + 4 has remainder 10 when divided by x − 2. Find k, then check whether x + 1 is a factor and explain what that would mean for the model's output.
Compare with the explanation
k = −1; x + 1 is not a factor; P(−1) = 4.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide