Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use division to reduce a polynomial modulo a quadratic.
- Justify the conclusion "x³+2x²+3x+4=(x+2)(x²+1)+(2x+2)" using the stated assumptions.
Before you start
Polynomial arithmetic.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Divide x³+2x²+3x+4 by x²+1.
Why this math matters
Use division to reduce a polynomial modulo a quadratic. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Coefficients lie in the real field.
- The divisor is nonzero.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Separate a polynomial quotient from its remainder
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Divide x³+2x²+3x+4 by x²+1.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Subtract x(x²+1), leaving 2x²+2x+4
Cancel the leading cubic term first.
Work through the mathematics
Subtract 2(x²+1), leaving 2x+2
The remaining degree is smaller than the divisor's degree.
Check the conclusion
x³+2x²+3x+4=(x+2)(x²+1)+(2x+2)
Expanding the right side verifies both the quotient and the remainder.
The result
x³+2x²+3x+4=(x+2)(x²+1)+(2x+2)
Expanding the right side verifies both the quotient and the remainder.
Common mistakes to catch
- A remainder is a polynomial, not always a number.
- The remainder theorem for x−a is a special case, not every divisor.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the remainder of x³ modulo x²+1?
Show a hint
Replace x² by −1 in the quotient ring.
Reveal answer and explanation
−x
x³=x(x²+1)−x.
Practice 2
Why must the remainder have degree below two?
Show a hint
Otherwise another leading-term cancellation is possible.
Reveal answer and explanation
To make the division result canonical
Allowing higher-degree remainders would make the quotient-remainder pair nonunique.
Take the idea with you
Reduce a long polynomial calculation to two coordinates in R[x]/(x²+1).
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Calculate in a finite field with four elements
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