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The real number that should be subtracted from the polynomial f(x) = 15x5 + 70x4 + 35x3 - 135x2 - 40x - 11 so that it is exactly divisible by 5x4 + 10x3 - 15x2 - 5x is __________.
- a)-12
- b)-11
- c)11
- d)12
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The real number that should be subtracted from the polynomial f(x) = 1...
On dividing, 15x5 + 70x4 + 35x3 - 135x2 - 40x - 11 by 5x4 + 10x3 - 15x2 - 5x
We get, 3x + 8 as quotient and remainder as -11.
So if we subtract -11 from 15x5 + 70x4+35x3 - 135x2 - 40x - 11 it will be exactly divisible by 5x4 + 10x3 - 15x2 - 5x.
We get, 3x + 8 as quotient and remainder as -11.
So if we subtract -11 from 15x5 + 70x4+35x3 - 135x2 - 40x - 11 it will be exactly divisible by 5x4 + 10x3 - 15x2 - 5x.
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The real number that should be subtracted from the polynomial f(x) = 1...
Given Polynomial: f(x) = 15x^5 - 70x^4 + 35x^3 - 135x^2 - 40x - 11
Divisor Polynomial: g(x) = 5x^4 + 10x^3 - 15x^2 - 5x
To find the real number that should be subtracted from f(x) to make it exactly divisible by g(x), we need to perform polynomial long division.
Polynomial Long Division:
Step 1: Divide the highest degree term of f(x) by the highest degree term of g(x).
15x^5 / 5x^4 = 3x
Step 2: Multiply g(x) by the quotient obtained in Step 1.
3x * (5x^4 + 10x^3 - 15x^2 - 5x) = 15x^5 + 30x^4 - 45x^3 - 15x^2
Step 3: Subtract the result obtained in Step 2 from f(x).
(15x^5 - 70x^4 + 35x^3 - 135x^2 - 40x - 11) - (15x^5 + 30x^4 - 45x^3 - 15x^2) = -100x^4 + 80x^3 - 120x^2 - 40x - 11
Step 4: Repeat Steps 1-3 with the new polynomial obtained in Step 3.
-100x^4 / 5x^4 = -20
-20 * (5x^4 + 10x^3 - 15x^2 - 5x) = -100x^4 - 200x^3 + 300x^2 + 100x
(-100x^4 + 80x^3 - 120x^2 - 40x - 11) - (-100x^4 - 200x^3 + 300x^2 + 100x) = 280x^3 - 420x^2 - 140x - 11
Step 5: Repeat Steps 1-3 with the new polynomial obtained in Step 4.
280x^3 / 5x^4 = 56/x
56/x * (5x^4 + 10x^3 - 15x^2 - 5x) = 280x^3 + 560x^2 - 840x - 280
(280x^3 - 420x^2 - 140x - 11) - (280x^3 + 560x^2 - 840x - 280) = -980x^2 + 700x - 291
Since the degree of the remainder (-980x^2 + 700x - 291) is less than the degree of g(x), we have reached the end of the polynomial long division.
Conclusion:
The remainder obtained after polynomial long division is -980x^2 + 700x - 291. This is the real number that should be subtracted from f(x) to make it exactly divisible by g(x). Therefore, the correct answer is option 'B
Divisor Polynomial: g(x) = 5x^4 + 10x^3 - 15x^2 - 5x
To find the real number that should be subtracted from f(x) to make it exactly divisible by g(x), we need to perform polynomial long division.
Polynomial Long Division:
Step 1: Divide the highest degree term of f(x) by the highest degree term of g(x).
15x^5 / 5x^4 = 3x
Step 2: Multiply g(x) by the quotient obtained in Step 1.
3x * (5x^4 + 10x^3 - 15x^2 - 5x) = 15x^5 + 30x^4 - 45x^3 - 15x^2
Step 3: Subtract the result obtained in Step 2 from f(x).
(15x^5 - 70x^4 + 35x^3 - 135x^2 - 40x - 11) - (15x^5 + 30x^4 - 45x^3 - 15x^2) = -100x^4 + 80x^3 - 120x^2 - 40x - 11
Step 4: Repeat Steps 1-3 with the new polynomial obtained in Step 3.
-100x^4 / 5x^4 = -20
-20 * (5x^4 + 10x^3 - 15x^2 - 5x) = -100x^4 - 200x^3 + 300x^2 + 100x
(-100x^4 + 80x^3 - 120x^2 - 40x - 11) - (-100x^4 - 200x^3 + 300x^2 + 100x) = 280x^3 - 420x^2 - 140x - 11
Step 5: Repeat Steps 1-3 with the new polynomial obtained in Step 4.
280x^3 / 5x^4 = 56/x
56/x * (5x^4 + 10x^3 - 15x^2 - 5x) = 280x^3 + 560x^2 - 840x - 280
(280x^3 - 420x^2 - 140x - 11) - (280x^3 + 560x^2 - 840x - 280) = -980x^2 + 700x - 291
Since the degree of the remainder (-980x^2 + 700x - 291) is less than the degree of g(x), we have reached the end of the polynomial long division.
Conclusion:
The remainder obtained after polynomial long division is -980x^2 + 700x - 291. This is the real number that should be subtracted from f(x) to make it exactly divisible by g(x). Therefore, the correct answer is option 'B
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The real number that should be subtracted from the polynomial f(x) = 15x5+ 70x4+ 35x3- 135x2- 40x - 11 so that it is exactly divisible by 5x4+ 10x3- 15x2- 5x is __________.a)-12b)-11c)11d)12Correct answer is option 'B'. Can you explain this answer?
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