If (p(x)=(x-c)^5(x+d)^2), where (c\neq -d), what are the distinct zeroes?
Answer and explanation
Correct answer: (c) and (-d)
The zeroes of a factored polynomial are found by setting each factor equal to zero. A factor raised to a power still contributes only one distinct zero; its exponent gives the multiplicity. This distinction is important because the question asks for distinct zeroes rather than all zeroes counted with repetition. The given inequality ensures that the two values do not coincide.
From \(x-c=0\), we obtain \(x=c\). From \(x+d=0\), we obtain \(x=-d\). The powers 5 and 2 mean that these zeroes have multiplicities 5 and 2, respectively, but the distinct list is only \(c\) and \(-d\). Since \(c\ne-d\), they are genuinely different. Therefore option A is correct; option C repeats \(-d\) unnecessarily.
Frequently asked questions
What is the correct answer to this question?
(c) and (-d)
Why is this the correct answer?
The zeroes of a factored polynomial are found by setting each factor equal to zero. A factor raised to a power still contributes only one distinct zero; its exponent gives the multiplicity. This distinction is important because the question asks for distinct zeroes rather than all zeroes counted with repetition. The given inequality ensures that the two values do not coincide.
From \(x-c=0\), we obtain \(x=c\). From \(x+d=0\), we obtain \(x=-d\). The powers 5 and 2 mean that these zeroes have multiplicities 5 and 2, respectively, but the distinct list is only \(c\) and \(-d\). Since \(c\ne-d\), they are genuinely different. Therefore option A is correct; option C repeats \(-d\) unnecessarily.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Geometrical meaning of the zeroes of a polynomial..
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