divide the polynomials with long division
(16b^4-1)/(2b+1)
To divide the polynomial (16b^4-1) by (2b+1) using long division, follow these steps:
Step 1: Divide the first term of the dividend by the first term of the divisor.
Divide 16b^4 by 2b to get 8b^3.
Step 2: Multiply the divisor by the quotient obtained in the previous step.
Multiply (2b+1) by 8b^3 to get 16b^4 + 8b^3.
Step 3: Subtract the obtained product from the dividend.
Subtract (16b^4 + 8b^3) from (16b^4 - 1) to get -8b^3 - 1.
Step 4: Bring down the next term from the dividend.
Bring down the -1 to get -8b^3 - 1.
Step 5: Repeat steps 1-4 with the new polynomial -8b^3 - 1.
Step 6: Divide the first term of the new polynomial (-8b^3) by the first term of the divisor (2b) to get -4b^2.
Step 7: Multiply the divisor by the new quotient (-4b^2) to get -8b^3 - 4b^2.
Step 8: Subtract the obtained product from the new polynomial.
Subtract (-8b^3 - 4b^2) from (-8b^3 - 1) to get 4b^2 - 1.
Step 9: Bring down the next term from the dividend.
Bring down the -1 to get 4b^2 - 1.
Step 10: Repeat steps 1-4 with the new polynomial 4b^2 - 1.
Step 11: Divide the first term of the new polynomial (4b^2) by the first term of the divisor (2b) to get 2b.
Step 12: Multiply the divisor by the new quotient (2b) to get 4b^2 + 2b.
Step 13: Subtract the obtained product from the new polynomial.
Subtract (4b^2 + 2b) from (4b^2 - 1) to get -2b - 1.
Step 14: Bring down the next term from the dividend.
Bring down the -1 to get -2b - 1.
Step 15: Repeat steps 1-4 with the new polynomial -2b - 1.
Step 16: Divide the first term of the new polynomial (-2b) by the first term of the divisor (2b) to get -1.
Step 17: Multiply the divisor by the new quotient (-1) to get -2b - 1.
Step 18: Subtract the obtained product from the new polynomial.
Subtract (-2b - 1) from (-2b - 1) to get 0.
Step 19: There is no more term left in the dividend, so the final quotient is 8b^3 - 4b^2 + 2b - 1.
Therefore, (16b^4 - 1) divided by (2b + 1) equals 8b^3 - 4b^2 + 2b - 1.