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Grade 11 · Polynomial roots

Polynomial roots: A repeated negative root

Polynomial roots: investigate a repeated negative root with first root a = -2; second root b = -2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Polynomial roots: A repeated negative root. Input x: -4. Product value: 4. Slope: -4. Symmetry line x: -2Factor signs and intercepts036-404yx → · labeled axes rescale to this model
The leading coefficient is one and inputs are real. The motion traces a fixed function rather than moving its roots. Tangent slopes are exact derivatives, and the plot rescales vertically for the selected roots.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Input x
-4
Product value
4
Slope
-4
Symmetry line x
-2

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From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Factored form locates the zeros before any expansion. Two different roots create two crossings, whereas coincident roots produce a repeated factor and a touching point. The moving input exposes how the signs of the two factors determine the graph's sign between and outside the roots. This investigation starts with First root a = -2; Second root b = -2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f(x)=(x−a)(x−b); f′(x)=2x−a−b

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Identify the two selected roots and predict the intervals where the factors have opposite signs. The starting case is “A repeated negative root.”

  2. STEP 2

    Follow the changing quantity

    Play the input from −4 to 4. Compare the moving point with the two intercepts and the symmetry line halfway between them.

  3. STEP 3

    Explain and test the result

    Set the roots equal, then separate them again. Explain why the repeated-root case has no negative interval.

Your turn to explain

Make a prediction. Test your reasoning.

Keep First root a = -2; Second root b = -2. Pause the timeline at 60%. Given input x = 0.8, calculate product value, slope, symmetry line x. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Substitute x=0.8: (0.8−(-2))(0.8−(-2))=7.84. Differentiating gives 2(0.8)−(-2)−(-2)=5.6. Results: Product value: 7.84; Slope: 5.6; Symmetry line x: -2. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Predict whether a polynomial crosses or touches an axis

Use root multiplicity to interpret local sign changes.