Number of Zeroes of a Polynomial from Graphs

 

❓ Question

Look at the graphs given below.
Each is the graph of:

$$y = p(x)$$

where $p(x)$ is a polynomial.

For each graph, find the number of zeroes of $p(x)$.

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🖼️ Solution Image

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✍️ Short Explanation

The number of zeroes of a polynomial is equal to the number of points where its graph:

• intersects the x-axis, or
• touches the x-axis 💯

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🔹 Step 1 — Graph (i)

The graph intersects the x-axis at:

$$1 \text{ point}$$

Therefore,

$$\boxed{1 \text{ zero}}$$

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🔹 Step 2 — Graph (ii)

The graph intersects the x-axis at:

$$2 \text{ points}$$

Therefore,

$$\boxed{2 \text{ zeroes}}$$

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🔹 Step 3 — Graph (iii)

The graph intersects the x-axis at:

$$3 \text{ points}$$

Therefore,

$$\boxed{3 \text{ zeroes}}$$

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🔹 Step 4 — Graph (iv)

The graph intersects the x-axis at:

$$1 \text{ point}$$

Therefore,

$$\boxed{1 \text{ zero}}$$

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🔹 Step 5 — Graph (v)

The graph touches the x-axis at:

$$1 \text{ point}$$

Therefore,

$$\boxed{1 \text{ zero}}$$

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🔹 Step 6 — Graph (vi)

The graph intersects the x-axis at:

$$4 \text{ points}$$

Therefore,

$$\boxed{4 \text{ zeroes}}$$

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✅ Final Answer

Graph	Number of Zeroes
(i)	1
(ii)	2
(iii)	3
(iv)	1
(v)	1
(vi)	4

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⭐ Key Insight

• Number of zeroes = number of x-axis intersections
• Touching the x-axis also counts as a zero

🧠 Memory Line:

X-axis ko jitni baar graph cut/touch karega, utne zeroes honge

📚 Related Topics

• Prove that 3 + 2√5 is Irrational
• Proof that √5 is Irrational
• Irrational Numbers Concept Using Contradiction Method