If p(x) = x³ − 4x² − 7x + 10 and (x − 1) is a factor, what is the remaining quadratic factor?
Answer and explanation
Correct answer: x² − 3x − 10
The governing idea is the factor theorem and polynomial division. Since (x − 1) is a factor, divide p(x) by (x − 1), or use synthetic division with 1. The coefficients 1, −4, −7, 10 give a quotient with coefficients 1, −3, −10 and remainder 0. Therefore p(x) = (x − 1)(x² − 3x − 10). Multiplication confirms this: x³ − 3x² − 10x − x² + 3x + 10 = x³ − 4x² − 7x + 10. Thus option A is correct. Option B has the wrong sign for the middle term, option C does not reproduce the cubic, and option D is only part of the original coefficient pattern, not a quadratic quotient.
Frequently asked questions
What is the correct answer to this question?
x² − 3x − 10
Why is this the correct answer?
The governing idea is the factor theorem and polynomial division. Since (x − 1) is a factor, divide p(x) by (x − 1), or use synthetic division with 1. The coefficients 1, −4, −7, 10 give a quotient with coefficients 1, −3, −10 and remainder 0. Therefore p(x) = (x − 1)(x² − 3x − 10). Multiplication confirms this: x³ − 3x² − 10x − x² + 3x + 10 = x³ − 4x² − 7x + 10. Thus option A is correct. Option B has the wrong sign for the middle term, option C does not reproduce the cubic, and option D is only part of the original coefficient pattern, not a quadratic quotient.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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