Divide using long division. $(4x^3-3x^2-2x+1)÷(x+1)$

Verified step by step guidance

1. Set up the long division by writing the dividend \(4x^3 - 3x^2 - 2x + 1\) under the division bar and the divisor \(x + 1\) outside the division bar.
2. Divide the leading term of the dividend, \$4x^3\(, by the leading term of the divisor, \)x\(, to get the first term of the quotient: \)4x^2$.
3. Multiply the entire divisor \(x + 1\) by \$4x^2$ to get \(4x^3 + 4x^2\), then subtract this from the dividend to find the new remainder.
4. Bring down the next term from the original dividend and repeat the process: divide the leading term of the new remainder by \(x\), multiply the divisor by this result, subtract, and continue.
5. Continue this process until all terms have been brought down and divided, resulting in the quotient and possibly a remainder expressed over the divisor.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting from the dividend, and repeating the process until the remainder has a lower degree than the divisor.

Leading Terms and Degree of Polynomials

The leading term of a polynomial is the term with the highest exponent, and the degree is the exponent of that term. Understanding these helps determine the first step in division and when to stop the process, as division continues until the remainder's degree is less than the divisor's degree.

Remainder and Quotient in Polynomial Division

In polynomial division, the quotient is the result of the division, and the remainder is what is left when the division cannot continue. The remainder must have a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.