The graph of a polynomial P(x) cuts the x-axis at 3 points and touches it at 2 other points. The number of zeroes of P(x) is (a) 1 (b) 2 (c) 3 (d) 5
Answers
The answer should be 2
not 5
The number of zeroes of P(x) is 5 if The graph of a polynomial P(x) cuts the x-axis at 3 points and touches it at 2 other points.
Given:
- The graph of a polynomial P(x)
- cuts the x-axis at 3 points
- touches it at 2 other points
To Find:
- The number of zeroes of P(x)
Multiplicity of Zeros of a polynomial
- For a zero b with odd multiplicity, the function changes sign at x= b and the graph crosses the x-axis.
- For a zero b with even multiplicity, the function does not change signs at x=b and the graph comes in contact with the -axis without crossing it.
The graph of a polynomial P(x) cuts the x-axis at 3 points hence,
at 3 points there are 3 zeroes with odd multiplicity
The graph of a polynomial P(x) touches it at 2 other points hence,
at 2 points there are 2 zeroes with even multiplicity
Total Number of zeroes = 3 + 2 = 5
Correct option is d ) 5
Minimum Degree of polynomial is 1 * 3 + 2 * 2 = 7
An example is shown and attached
P(x) = x(x - 1)²(x + 1)²(x - 2)(x + 2)
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