If (p(x)=(x+a)^3(x-b)^2), where (a\neq -b), what are the distinct zeroes?
Answer and explanation
Correct answer: (-a) and (b)
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In a product, the polynomial becomes zero whenever at least one factor becomes zero. The exponents show multiplicity, or how many times a zero is repeated, but they do not create different zero values. The condition ensures that the two values obtained here are distinct.
Set the first factor equal to zero: \(x+a=0\), so \(x=-a\). Set the second factor equal to zero: \(x-b=0\), so \(x=b\). The powers 3 and 2 indicate repeated zeroes, but the distinct zeroes are counted only once each. Since \(a\ne-b\), these values are different. Hence option A, \(-a\) and \(b\), is correct.
Frequently asked questions
What is the correct answer to this question?
(-a) and (b)
Why is this the correct answer?
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In a product, the polynomial becomes zero whenever at least one factor becomes zero. The exponents show multiplicity, or how many times a zero is repeated, but they do not create different zero values. The condition ensures that the two values obtained here are distinct.
Set the first factor equal to zero: \(x+a=0\), so \(x=-a\). Set the second factor equal to zero: \(x-b=0\), so \(x=b\). The powers 3 and 2 indicate repeated zeroes, but the distinct zeroes are counted only once each. Since \(a\ne-b\), these values are different. Hence option A, \(-a\) and \(b\), is correct.
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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