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What must be subtracted from 8x4 + 14x3 -2x2 + 7x - 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x - 2?
- a)4x -10
- b)14x - 10
- c)2x - 10
- d)14x + 10
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
What must be subtracted from 8x4 + 14x3 -2x2 + 7x - 8 so that the resu...
To find what must be subtracted from the polynomial so that the resulting polynomial is exactly divisible by another polynomial, we need to use the concept of polynomial division.
The given polynomial is:
8x^4 + 14x^3 - 2x^2 + 7x - 8
The polynomial we want to divide by is:
4x^2 + 3x - 2
Step 1: Divide the leading term
We will divide the leading term of the given polynomial by the leading term of the divisor.
Leading term of the given polynomial: 8x^4
Leading term of the divisor: 4x^2
8x^4 / 4x^2 = 2x^2
Step 2: Multiply the divisor by the quotient obtained in the previous step
Multiply the divisor by the quotient obtained in Step 1:
2x^2 * (4x^2 + 3x - 2) = 8x^4 + 6x^3 - 4x^2
Step 3: Subtract the result obtained in Step 2 from the given polynomial
Subtract the result obtained in Step 2 from the given polynomial:
(8x^4 + 14x^3 - 2x^2 + 7x - 8) - (8x^4 + 6x^3 - 4x^2)
8x^4 + 14x^3 - 2x^2 + 7x - 8 - 8x^4 - 6x^3 + 4x^2 = 8x^4 - 8x^4 + 14x^3 - 6x^3 - 2x^2 + 4x^2 + 7x - 8 = 8x^3 + 2x^2 + 7x - 8
Step 4: Repeat the process
Repeat Steps 1-3 with the new polynomial obtained in Step 3 until we are left with a polynomial that is completely divisible by the divisor.
2x^2 / 4x^2 = 1/2
1/2 * (4x^2 + 3x - 2) = 2x^2 + (3/2)x - 1
(8x^3 + 2x^2 + 7x - 8) - (2x^2 + (3/2)x - 1)
8x^3 + 2x^2 + 7x - 8 - 2x^2 - (3/2)x + 1 = 8x^3 - 2x^2 + (7 - (3/2))x - 7
Step 5: Final result
The resulting polynomial is 8x^3 - 2x^2 + (7 - (3/2))x - 7.
To make this polynomial exactly divisible by the given divisor, we need to subtract the remainder from the polynomial.
Remainder = 7 - (3/2)x - 7 = -(3/2)x
Therefore, the polynomial that must be subtracted is -(3/2)x.
Conclusion:
The correct option is
The given polynomial is:
8x^4 + 14x^3 - 2x^2 + 7x - 8
The polynomial we want to divide by is:
4x^2 + 3x - 2
Step 1: Divide the leading term
We will divide the leading term of the given polynomial by the leading term of the divisor.
Leading term of the given polynomial: 8x^4
Leading term of the divisor: 4x^2
8x^4 / 4x^2 = 2x^2
Step 2: Multiply the divisor by the quotient obtained in the previous step
Multiply the divisor by the quotient obtained in Step 1:
2x^2 * (4x^2 + 3x - 2) = 8x^4 + 6x^3 - 4x^2
Step 3: Subtract the result obtained in Step 2 from the given polynomial
Subtract the result obtained in Step 2 from the given polynomial:
(8x^4 + 14x^3 - 2x^2 + 7x - 8) - (8x^4 + 6x^3 - 4x^2)
8x^4 + 14x^3 - 2x^2 + 7x - 8 - 8x^4 - 6x^3 + 4x^2 = 8x^4 - 8x^4 + 14x^3 - 6x^3 - 2x^2 + 4x^2 + 7x - 8 = 8x^3 + 2x^2 + 7x - 8
Step 4: Repeat the process
Repeat Steps 1-3 with the new polynomial obtained in Step 3 until we are left with a polynomial that is completely divisible by the divisor.
2x^2 / 4x^2 = 1/2
1/2 * (4x^2 + 3x - 2) = 2x^2 + (3/2)x - 1
(8x^3 + 2x^2 + 7x - 8) - (2x^2 + (3/2)x - 1)
8x^3 + 2x^2 + 7x - 8 - 2x^2 - (3/2)x + 1 = 8x^3 - 2x^2 + (7 - (3/2))x - 7
Step 5: Final result
The resulting polynomial is 8x^3 - 2x^2 + (7 - (3/2))x - 7.
To make this polynomial exactly divisible by the given divisor, we need to subtract the remainder from the polynomial.
Remainder = 7 - (3/2)x - 7 = -(3/2)x
Therefore, the polynomial that must be subtracted is -(3/2)x.
Conclusion:
The correct option is
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Community Answer
What must be subtracted from 8x4 + 14x3 -2x2 + 7x - 8 so that the resu...
Here 4x2 + 3x - 2) 8x4 + 14x3 - 2x2 + 7x - 8 (2x2 + 2x -1
∴ 14x - 10 must be subtracted.
∴ 14x - 10 must be subtracted.
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What must be subtracted from 8x4 + 14x3 -2x2 + 7x - 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x - 2?a)4x -10b)14x - 10c)2x - 10d)14x + 10Correct answer is option 'B'. Can you explain this answer?
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