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What will be the value of a and b if the polynomial f(x) = 30x4 - 50x3 + 109x2 - 23x + 25, when divided by 3x2 - 5x + 10, gives 10x2 + 3 as quotient and ax + b as remainder?
- a)a = 8, b = 5
- b)a = -8, b = 5
- c)a = 8, b = -5
- d)a = -8, b = -5
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+...
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 30x4 - 50x3 + 109x2 - 23x + 25 = (10x2 + 3)(3x2 - 5x + 10) + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 = 30x4 - 50x3 + 109x2 - 15x + 30 + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 - (30x4 - 50x3 + 109x2 - 15x + 30) = ax + b
-23x + 25 + 15x - 30 = ax + b
-8x - 5 = ax + b
∴ a = -8, b = -5
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 30x4 - 50x3 + 109x2 - 23x + 25 = (10x2 + 3)(3x2 - 5x + 10) + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 = 30x4 - 50x3 + 109x2 - 15x + 30 + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 - (30x4 - 50x3 + 109x2 - 15x + 30) = ax + b
-23x + 25 + 15x - 30 = ax + b
-8x - 5 = ax + b
∴ a = -8, b = -5
Most Upvoted Answer
What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+...
To find the values of a and b, we need to perform polynomial long division. The dividend is f(x) = 30x^4 - 50x^3 + 109x^2 - 23x + 25, and the divisor is g(x) = 3x^2 - 5x + 10. The quotient is given as Q(x) = 10x^2 + 3, and the remainder is supposed to be in the form R(x) = ax + b.
1. Polynomial Division:
Let's perform the polynomial long division to find the quotient and remainder.
```
10x^2 + 3
_________________________
3x^2 - 5x + 10 | 30x^4 - 50x^3 + 109x^2 - 23x + 25
- (30x^4 - 50x^3 + 100x^2)
_________________________
9x^2 - 23x + 25
- (9x^2 - 15x + 30)
_________________________
-8x - 5
```
The quotient is 10x^2 + 3, and the remainder is -8x - 5.
2. Equating Quotient and Remainder:
We can equate the quotient and remainder to their respective forms to find the values of a and b.
```
Quotient: 10x^2 + 3
Remainder: ax + b
```
Equating the coefficients of the corresponding terms, we get:
```
10x^2 = 0x^2 (from quotient)
a = -8 (from remainder)
3 = 0 (from quotient)
b = -5 (from remainder)
```
3. Conclusion:
From the equating coefficients, we found that a = -8 and b = -5. Therefore, the correct answer is option D: a = -8, b = -5.
1. Polynomial Division:
Let's perform the polynomial long division to find the quotient and remainder.
```
10x^2 + 3
_________________________
3x^2 - 5x + 10 | 30x^4 - 50x^3 + 109x^2 - 23x + 25
- (30x^4 - 50x^3 + 100x^2)
_________________________
9x^2 - 23x + 25
- (9x^2 - 15x + 30)
_________________________
-8x - 5
```
The quotient is 10x^2 + 3, and the remainder is -8x - 5.
2. Equating Quotient and Remainder:
We can equate the quotient and remainder to their respective forms to find the values of a and b.
```
Quotient: 10x^2 + 3
Remainder: ax + b
```
Equating the coefficients of the corresponding terms, we get:
```
10x^2 = 0x^2 (from quotient)
a = -8 (from remainder)
3 = 0 (from quotient)
b = -5 (from remainder)
```
3. Conclusion:
From the equating coefficients, we found that a = -8 and b = -5. Therefore, the correct answer is option D: a = -8, b = -5.
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Community Answer
What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+...
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 30x4 - 50x3 + 109x2 - 23x + 25 = (10x2 + 3)(3x2 - 5x + 10) + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 = 30x4 - 50x3 + 109x2 - 15x + 30 + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 - (30x4 - 50x3 + 109x2 - 15x + 30) = ax + b
-23x + 25 + 15x - 30 = ax + b
-8x - 5 = ax + b
∴ a = -8, b = -5
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 30x4 - 50x3 + 109x2 - 23x + 25 = (10x2 + 3)(3x2 - 5x + 10) + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 = 30x4 - 50x3 + 109x2 - 15x + 30 + ax + b
30x4 - 50x3 + 109x2 - 23x + 25 - (30x4 - 50x3 + 109x2 - 15x + 30) = ax + b
-23x + 25 + 15x - 30 = ax + b
-8x - 5 = ax + b
∴ a = -8, b = -5
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What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer?
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What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer?.
What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the value of a and b if the polynomial f(x) = 30x4- 50x3+ 109x2- 23x + 25, when divided by 3x2- 5x + 10, gives 10x2+ 3 as quotient and ax + b as remainder?a)a = 8, b = 5b)a = -8, b = 5c)a = 8, b = -5d)a = -8, b = -5Correct answer is option 'D'. Can you explain this answer?.
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