In Exercises 27–29, divide using long division. (4x^3 - 3x^2 - 2x + 1) ÷ (x + 1)

Verified step by step guidance

1. Identify the dividend \(4x^3 - 3x^2 - 2x + 1\) and the divisor \(x + 1\).
2. Divide the first term of the dividend \(4x^3\) by the first term of the divisor \(x\) to get the first term of the quotient \(4x^2\).
3. Multiply the entire divisor \(x + 1\) by the first term of the quotient \(4x^2\) and subtract the result from the dividend.
4. Bring down the next term from the dividend to form a new polynomial and repeat the division process with the new polynomial.
5. Continue this process until all terms of the dividend have been divided, ensuring to subtract and bring down terms as needed.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves a process similar to numerical long division, where you divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by this result, and subtract it from the original polynomial. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the polynomial 4x^3 - 3x^2 - 2x + 1, the leading coefficient is 4. This value is crucial in long division as it determines how the first term of the quotient is calculated, influencing the subsequent steps of the division process.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem is useful for checking the results of polynomial long division, as it allows you to quickly verify the remainder by substituting the value of c into the original polynomial.