Q1
The graphs of \(y = p(x)\) are given below. Find the number of zeroes of \(p(x)\) in each case.
Concept Before Solving
- A zero of a polynomial is a value of \(x\) for which \(p(x)=0\).
- Graphically, zeroes are the points where the graph intersects the \(x\)-axis.
- If graph cuts (crosses) the \(x\)-axis → simple zero.
- If graph just touches the \(x\)-axis → repeated zero (even multiplicity).
Solution Roadmap
- Step 1: Observe each graph carefully.
- Step 2: Count how many times graph meets \(x\)-axis.
- Step 3: Include both crossing and touching points.
- Step 4: That count = number of zeroes.
-
Figure (i):
The graph is a horizontal line parallel to the \(x\)-axis and does not intersect it.
Step-wise:
• No intersection with \(x\)-axis
• No solution to \(p(x)=0\)
Therefore, number of zeroes = 0 -
Figure (ii):
The graph intersects the \(x\)-axis at exactly one point.
Step-wise:
• Curve crosses axis once
• Only one value of \(x\) makes \(p(x)=0\)
Therefore, number of zeroes = 1 -
Figure (iii):
The graph cuts the \(x\)-axis three times.
Step-wise:
• First intersection (left)
• Second intersection (middle)
• Third intersection (right)
Therefore, number of zeroes = 3 -
Figure (iv):
The graph intersects the \(x\)-axis at two distinct points.
Step-wise:
• One intersection on left
• One intersection on right
Therefore, number of zeroes = 2 -
Figure (v):
The graph cuts the \(x\)-axis four times.
Step-wise:
• Four distinct intersections observed
• Each gives one zero
Therefore, number of zeroes = 4 -
Figure (vi):
The graph intersects once and touches twice.
Step-wise:
• One crossing → 1 zero
• Two touching points → 2 zeros
Total zeroes = \(1 + 2 = 3\)
Therefore, number of zeroes = 3
Key Observations (Very Important)
- Number of zeroes = number of intersections with \(x\)-axis.
- Touching point also counts as zero.
- Graph may have 0, 1, 2, 3... zeroes depending on shape.
Exam Significance
- Frequently asked in CBSE Board Exams (1–2 marks direct question).
- Forms base of graphical interpretation of polynomials.
- Important for JEE Foundation & NDA level problems.
- Helps in understanding roots, multiplicity, and curve behavior.