Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Use root multiplicity to interpret local sign changes.
- Justify the conclusion "The graph touches at x=1 and crosses at x=−2" using the stated assumptions.
Before you start
Factored polynomials and sign reasoning.
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
For p(x)=(x−1)²(x+2), describe behavior at its real zeros.
Why this math matters
Use root multiplicity to interpret local sign changes. 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 behavior is considered in a small neighborhood of each isolated root.
- All displayed factors have nonzero leading coefficient.
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.
Predict whether a polynomial crosses or touches an axis
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For p(x)=(x−1)²(x+2), describe behavior at its real zeros.
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
Zeros are x=1 with multiplicity two and x=−2 with multiplicity one
Exponents record how often each linear factor occurs.
Work through the mathematics
Near one, (x−1)² stays nonnegative while x+2 stays positive
The sign does not reverse at the double root.
Check the conclusion
The graph touches at x=1 and crosses at x=−2
The simple factor at −2 changes sign while the other factor stays positive.
The result
The graph touches at x=1 and crosses at x=−2
The simple factor at −2 changes sign while the other factor stays positive.
Common mistakes to catch
- Distinct roots and roots counted with multiplicity are different counts.
- Even multiplicity touching does not by itself determine global maxima or minima elsewhere.
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 happens at a triple root?
Show a hint
An odd power changes sign.
Reveal answer and explanation
The graph crosses with a flattened local shape
Odd multiplicity gives a sign change, while higher multiplicity can reduce local steepness.
Practice 2
How many roots are counted with multiplicity here?
Show a hint
Add the factor exponents.
Reveal answer and explanation
Three
Two plus one matches the polynomial's degree.
Take the idea with you
Read a polynomial's zero behavior from factors before drawing its full graph.
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: Expose a rational graph's shifted reciprocal structure
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