(X`2-7x+10x`3+1) ÷(2X-1)

To divide the polynomial (x^2 - 7x + 10x^3 + 1) by (2x - 1), we can use polynomial long division. Here's how you can do it step by step:

Step 1: Arrange the terms in descending order.
The polynomial (x^2 - 7x + 10x^3 + 1) can be rewritten as 10x^3 + x^2 - 7x + 1.

Step 2: Divide the first term of the dividend (10x^3) by the first term of the divisor (2x) to get the first term of the quotient.
The quotient's first term is (10x^3 / 2x) = 5x^2.

Step 3: Multiply the entire divisor (2x - 1) by the first term of the quotient (5x^2) and subtract the result from the dividend.
Multiplying (5x^2) by (2x - 1) gives you (10x^3 - 5x^2).
Subtracting (10x^3 - 5x^2) from (10x^3 + x^2 - 7x + 1) results in (x^2 - 7x + 1).

Step 4: Bring down the next term from the dividend.
The next term in the dividend is -7x, so bring it down.

Step 5: Divide the new polynomial (x^2 - 7x + 1) by the divisor (2x - 1) to get the second term of the quotient.
The quotient's second term is (x / 2) = 0.5x.

Step 6: Multiply the entire divisor (2x - 1) by the second term of the quotient (0.5x) and subtract the result from the current polynomial.
Multiplying (0.5x) by (2x - 1) gives you (x^2 - 0.5x).
Subtracting (x^2 - 0.5x) from (x^2 - 7x + 1) gives you (-6.5x + 1).

Step 7: Bring down the next term from the dividend.
The next term in the dividend is 1, so bring it down.

Step 8: Divide the new polynomial (-6.5x + 1) by the divisor (2x - 1) to get the last term of the quotient.
The last term of the quotient is (-6.5 / 2) = -3.25.

Step 9: Multiply the entire divisor (2x - 1) by the last term of the quotient (-3.25) and subtract the result from the current polynomial.
Multiplying (-3.25) by (2x - 1) gives you (-6.5x + 3.25).
Subtracting (-6.5x + 3.25) from (-6.5x + 1) gives you (-1.75).

Step 10: There are no more terms in the dividend to bring down, so the division is complete.

The quotient is 5x^2 + 0.5x - 3.25, and the remainder is -1.75.