Final answer:
To find the quotient of (4x^4-x^2-2x+1) ÷ (2x-3), polynomial long division or synthetic division can be used. Both methods result in the quotient being 2x^3 + 3x^2 + 7x + 7.
Step-by-step explanation:
The question involves polynomial long division and synthetic division.
To find the quotient of (4x^4-x^2-2x+1) ÷ (2x-3), we can first use polynomial long division:
1. We divide the first term of the numerator (4x^4) by the first term of the denominator (2x) which gives us 2x^3.
2. We multiply the entire denominator (2x-3) by this newly found term (2x^3) and subtract it from the polynomial.
3. We continue this process with the new polynomial until we cannot divide any further, giving us the quotient.
The same quotient can be found using synthetic division with less writing and operations:
1. We use the reciprocal of the root of the denominator, which would be 3/2 for (2x-3 = 0).
2. We write the coefficients of the polynomial in descending order and apply synthetic division rules.
3. The result gives us the coefficients of the quotient polynomial.
For both methods in this case, the quotient is 2x^3 + 3x^2 + 7x + 7.