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The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) is
- a)x(x + 1)2 (x - 1)
- b)x(x - 1)2 (x + 1)
- c)(x - 1)2 (x + 1)
- d)x2 (x + 1) (x - 1)2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The polynomial f (x) has a double root -1, single roots 1 and 0. Then ...
The Polynomial f(x) with Double Root -1, Single Roots 1 and 0
To determine the correct polynomial f(x) with the given roots, let's analyze the information provided.
Given:
- Double root at -1
- Single roots at 1 and 0
Understanding the Roots of a Polynomial:
- The roots of a polynomial are the values of x that make the polynomial equal to zero.
- A double root means that a particular value of x occurs twice in the polynomial.
- Single roots mean that a particular value of x occurs only once in the polynomial.
Determining the Polynomial:
- Since the polynomial has a double root at -1, we know that (x + 1) is a factor of the polynomial.
- Since the polynomial has single roots at 1 and 0, we know that (x - 1) and (x - 0) (which simplifies to just x) are also factors of the polynomial.
Using the Factors to Determine the Polynomial:
- We can multiply these factors together to find the polynomial: (x + 1)(x - 1)(x - 0).
- Simplifying this expression, we get f(x) = (x + 1)(x - 1)(x).
- Expanding further, we have f(x) = (x^2 - 1)(x).
- Multiplying again, we get f(x) = x^3 - x.
Comparing the Polynomial with the Given Options:
- Let's compare the polynomial we obtained (f(x) = x^3 - x) with the given options:
a) (x - 1)^2 (x + 1)
b) x(x - 1)^2 (x + 1)
c) x(x - 1)^2 (x - 1)
d) x^2 (x - 1) (x + 1)
Option A:
- (x - 1)^2 (x + 1) = (x^2 - 2x + 1)(x + 1) = x^3 - x^2 - x + 1
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option B:
- x(x - 1)^2 (x + 1) = x(x^2 - 2x + 1)(x + 1) = x^4 - 2x^3 + x^2 + x^2 - 2x + x = x^4 - 2x^3 + 2x^2 - x
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option C:
- x(x - 1)^2 (x - 1) = x(x^2 - 2x + 1)(x - 1) = x^4 - 2x^3 + x^2 - x^3 + 2x^2 - x = x^4 - 3x^3 + 3x^2 - x
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option D
To determine the correct polynomial f(x) with the given roots, let's analyze the information provided.
Given:
- Double root at -1
- Single roots at 1 and 0
Understanding the Roots of a Polynomial:
- The roots of a polynomial are the values of x that make the polynomial equal to zero.
- A double root means that a particular value of x occurs twice in the polynomial.
- Single roots mean that a particular value of x occurs only once in the polynomial.
Determining the Polynomial:
- Since the polynomial has a double root at -1, we know that (x + 1) is a factor of the polynomial.
- Since the polynomial has single roots at 1 and 0, we know that (x - 1) and (x - 0) (which simplifies to just x) are also factors of the polynomial.
Using the Factors to Determine the Polynomial:
- We can multiply these factors together to find the polynomial: (x + 1)(x - 1)(x - 0).
- Simplifying this expression, we get f(x) = (x + 1)(x - 1)(x).
- Expanding further, we have f(x) = (x^2 - 1)(x).
- Multiplying again, we get f(x) = x^3 - x.
Comparing the Polynomial with the Given Options:
- Let's compare the polynomial we obtained (f(x) = x^3 - x) with the given options:
a) (x - 1)^2 (x + 1)
b) x(x - 1)^2 (x + 1)
c) x(x - 1)^2 (x - 1)
d) x^2 (x - 1) (x + 1)
Option A:
- (x - 1)^2 (x + 1) = (x^2 - 2x + 1)(x + 1) = x^3 - x^2 - x + 1
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option B:
- x(x - 1)^2 (x + 1) = x(x^2 - 2x + 1)(x + 1) = x^4 - 2x^3 + x^2 + x^2 - 2x + x = x^4 - 2x^3 + 2x^2 - x
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option C:
- x(x - 1)^2 (x - 1) = x(x^2 - 2x + 1)(x - 1) = x^4 - 2x^3 + x^2 - x^3 + 2x^2 - x = x^4 - 3x^3 + 3x^2 - x
- This does not match the polynomial we obtained (f(x) = x^3 - x).
Option D
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Community Answer
The polynomial f (x) has a double root -1, single roots 1 and 0. Then ...
We know that, polynomial consists of several terms.
So, the given polynomial has double root which means 2nd degree, single root means single polynomial, no degree and 0.
So, the polynomial is (x - 1)2 (x + 1).
So, the given polynomial has double root which means 2nd degree, single root means single polynomial, no degree and 0.
So, the polynomial is (x - 1)2 (x + 1).
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The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer? for CTET & State TET 2026 is part of CTET & State TET preparation. The Question and answers have been prepared according to the CTET & State TET exam syllabus. Information about The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CTET & State TET 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer?.
The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer? for CTET & State TET 2026 is part of CTET & State TET preparation. The Question and answers have been prepared according to the CTET & State TET exam syllabus. Information about The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CTET & State TET 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomial f (x) has a double root -1, single roots 1 and 0. Then f (x) isa)x(x + 1)2 (x - 1)b)x(x - 1)2 (x + 1)c)(x - 1)2 (x + 1)d)x2 (x + 1) (x - 1)2Correct answer is option 'A'. Can you explain this answer?.
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