If (p(x)=(x+1)(x-4)^3), what are the distinct zeroes?
Answer and explanation
Correct answer: (-1) and (4)
To find the zeroes of a factored polynomial, set each factor equal to zero. A product is zero whenever at least one of its factors is zero. In this expression, the factor \(x+1\) gives one zero, while the factor \((x-4)^3\) gives another zero. The exponent 3 shows that the second zero occurs repeatedly, but it does not create three different zeroes.
Solving \(x+1=0\) gives \(x=-1\). Solving \(x-4=0\) gives \(x=4\). Thus the zeroes are \(-1\) and \(4\), and the distinct zeroes are exactly these two values. Option C lists the repeated value three times, so it describes multiplicity rather than distinct zeroes. Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
(-1) and (4)
Why is this the correct answer?
To find the zeroes of a factored polynomial, set each factor equal to zero. A product is zero whenever at least one of its factors is zero. In this expression, the factor \(x+1\) gives one zero, while the factor \((x-4)^3\) gives another zero. The exponent 3 shows that the second zero occurs repeatedly, but it does not create three different zeroes.
Solving \(x+1=0\) gives \(x=-1\). Solving \(x-4=0\) gives \(x=4\). Thus the zeroes are \(-1\) and \(4\), and the distinct zeroes are exactly these two values. Option C lists the repeated value three times, so it describes multiplicity rather than distinct zeroes. Therefore option A 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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