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.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Apply the factor theorem to an unknown polynomial parameter.
- Justify the conclusion "p(x)=x³−x−6=(x−2)(x²+2x+3)" using the stated assumptions.
Before you start
Polynomial substitution and factorization.
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
If x−2 is a factor of p(x)=x³+kx−6, find k.
Why this math matters
Apply the factor theorem to an unknown polynomial parameter. 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:
- The polynomial has the displayed form.
- Coefficients and roots are real unless otherwise stated.
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.
Use a known root to determine a coefficient
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If x−2 is a factor of p(x)=x³+kx−6, find k.
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
A factor x−2 requires p(2)=0
The remainder on division by a linear factor equals the value at its root.
Work through the mathematics
8+2k−6=0 ⇒ k=−1
Substitution turns the factor requirement into a linear equation.
Check the conclusion
p(x)=x³−x−6=(x−2)(x²+2x+3)
Expansion verifies the factor and the recovered coefficient.
The result
p(x)=x³−x−6=(x−2)(x²+2x+3)
Expansion verifies the factor and the recovered coefficient.
Common mistakes to catch
- For x+a, evaluate at −a.
- Knowing one factor does not determine every remaining root.
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
Is x−1 also a factor?
Show a hint
Evaluate p(1).
Reveal answer and explanation
No
p(1)=−6, so the remainder is nonzero.
Practice 2
What is the remainder when p is divided by x+1?
Show a hint
Evaluate at −1.
Reveal answer and explanation
−6
p(−1)=−1+1−6.
Take the idea with you
Use a required zero response to calibrate a polynomial model's free coefficient.
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: Solve a rational inequality by intervals
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