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Class 10 Zeroes of a Polynomial — practice questions with answers

Algebra — Polynomials · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.

1. What is the value of the polynomial p(x) = x - 3 at x = 3?
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Answer: D) 0
p(3) = 3 - 3 = 0. Since p(3) = 0, x = 3 is a zero of the polynomial.
2. The graph of a polynomial touches the x-axis at exactly one point and goes back up. How many zeroes does it have at that point (counting multiplicity)?
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Answer: C) 2
When a graph just touches the x-axis (does not cross), that point is a repeated zero; it counts as 2 equal zeroes.
3. The constant polynomial p(x) = 5 has how many zeroes?
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Answer: A) 0
p(x) = 5 is never equal to 0, so it has no zeroes. Number of zeroes = 0.
4. Construct a quadratic polynomial whose zeroes are 2 and -7. Which polynomial below is correct (with leading coefficient 1)?
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Answer: A) x² + 5x - 14
Sum = 2 + (-7) = -5; product = 2(-7) = -14. Polynomial = x² - (sum)x + product = x² + 5x - 14.
5. A polynomial p(x) of degree 5 has exactly 3 distinct real zeroes, and its graph just touches (does not cross) the x-axis at one of them.
What is the smallest total number of real zeroes counted with multiplicity that this graph can show?
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Answer: A) 4
Two zeroes are simple crossings (count 1 each = 2). One zero is a touch, so it is repeated at least twice (count 2). Minimum total counted with multiplicity = 1 + 1 + 2 = 4 (the 5th would come from further multiplicity or complex zeroes).
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