Divide using long division. State the quotient, and the remainder, r(x). (x⁴+2x³−4x²−5x−6)/(x²+x−2)

Verified step by step guidance

1. Identify the dividend and divisor: The dividend is \(x^{4} + 2x^{3} - 4x^{2} - 5x - 6\) and the divisor is \(x^{2} + x - 2\).
2. Set up the long division by writing the dividend under the division bar and the divisor outside.
3. Divide the leading term of the dividend \(x^{4}\) by the leading term of the divisor \(x^{2}\) to find the first term of the quotient: \(\frac{x^{4}}{x^{2}} = x^{2}\).
4. Multiply the entire divisor \(x^{2} + x - 2\) by \(x^{2}\) and subtract the result from the dividend to find the new polynomial to divide.
5. Repeat the process: divide the leading term of the new polynomial by \(x^{2}\), multiply the divisor by this term, subtract, and continue until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, 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, and repeating until the degree of the remainder is less than the divisor.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding degrees is essential in division because the process continues until the remainder's degree is less than the divisor's degree, indicating the division is complete.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the result of the division, and the remainder is what is left over. The remainder must have a degree less than the divisor. Expressing the division as dividend = divisor × quotient + remainder helps verify the result.