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If α is a zero of the polynomial f(x), then the divisor of f(x) will be _________
- a)x < α
- b)x - α
- c)x > α
- d)x + α
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If α is a zero of the polynomial f(x), then the divisor of f(x) ...
If α is a zero of the polynomial f(x).
The divisor will be x - α.
For example, if 5 is a zero of a polynomial f(x), then its divisor will be x - 5.
The divisor will be x - α.
For example, if 5 is a zero of a polynomial f(x), then its divisor will be x - 5.
Most Upvoted Answer
If α is a zero of the polynomial f(x), then the divisor of f(x) ...
If α is a zero of the polynomial f(x).
The divisor will be x - α.
For example, if 5 is a zero of a polynomial f(x), then its divisor will be x - 5.
The divisor will be x - α.
For example, if 5 is a zero of a polynomial f(x), then its divisor will be x - 5.
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If α is a zero of the polynomial f(x), then the divisor of f(x) ...
Understanding Polynomial Zeros
When we talk about a polynomial f(x), a zero (or root) of the polynomial is a value of x (denoted as α) that makes the polynomial equal to zero, i.e., f(α) = 0. This is crucial in polynomial division and factorization.
Divisibility by Factors
If α is a zero of f(x), it implies that (x - α) is a factor of f(x). This relationship is based on the Factor Theorem, which states:
- If f(α) = 0, then (x - α) divides f(x) without a remainder.
Examining the Options
Given the options:
- a) x < />
- b) x - α
- c) x > α
- d) x + α
Only option b) (x - α) correctly represents the factor associated with the zero α.
Why Other Options are Incorrect
- a) x < α:="" this="" option="" does="" not="" represent="" a="" factor="" of="" the="" polynomial;="" it="" simply="" describes="" a="" range="" of="" />
- c) x > α: Similar to a), this option describes another range of values and does not correspond to a polynomial factor.
- d) x + α: This option suggests a different relationship and does not indicate that α is a zero of f(x).
Conclusion
Thus, the correct divisor of f(x) when α is a zero is indeed:
Option b) x - α
This reflects the fundamental relationship between zeros of polynomials and their factors.
When we talk about a polynomial f(x), a zero (or root) of the polynomial is a value of x (denoted as α) that makes the polynomial equal to zero, i.e., f(α) = 0. This is crucial in polynomial division and factorization.
Divisibility by Factors
If α is a zero of f(x), it implies that (x - α) is a factor of f(x). This relationship is based on the Factor Theorem, which states:
- If f(α) = 0, then (x - α) divides f(x) without a remainder.
Examining the Options
Given the options:
- a) x < />
- b) x - α
- c) x > α
- d) x + α
Only option b) (x - α) correctly represents the factor associated with the zero α.
Why Other Options are Incorrect
- a) x < α:="" this="" option="" does="" not="" represent="" a="" factor="" of="" the="" polynomial;="" it="" simply="" describes="" a="" range="" of="" />
- c) x > α: Similar to a), this option describes another range of values and does not correspond to a polynomial factor.
- d) x + α: This option suggests a different relationship and does not indicate that α is a zero of f(x).
Conclusion
Thus, the correct divisor of f(x) when α is a zero is indeed:
Option b) x - α
This reflects the fundamental relationship between zeros of polynomials and their factors.
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If α is a zero of the polynomial f(x), then the divisor of f(x) will be _________a)x < αb)x - αc)x > αd)x + αCorrect answer is option 'B'. Can you explain this answer?
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