Long Division Method for Polynomial Division

When dividing polynomials, we use the long division method, which is similar to the division of integers. The steps involved in the long division method for polynomial division are as follows:

1. Arrange the dividend and divisor in standard form with the terms arranged in descending order of degree.
2. Divide the first term of the dividend by the first term of the divisor.
3. Write the quotient above the dividend and multiply the quotient by the divisor.
4. Subtract the product from the dividend.
5. Bring down the next term of the dividend and repeat steps 2 to 4 until the degree of the remainder is less than the degree of the divisor.

Dividing 3x^4-4x^3-3x-1 by x-2

To divide 3x^4-4x^3-3x-1 by x-2, we follow the steps of the long division method as outlined above:



Therefore, the quotient is 3x^3 + 2x^2 + 7x + 14, and the remainder is 5x - 1.

Explanation

- We begin by arranging the dividend and divisor in standard form with the terms arranged in descending order of degree. In this case, the dividend is 3x^4 - 4x^3 - 3x - 1, and the divisor is x - 2.
- We divide the first term of the dividend, which is 3x^4, by the first term of the divisor, which is x. This gives us a quotient of 3x^3.
- We write the quotient above the dividend, as shown in the long division format. We then multiply the quotient by the divisor, which gives us 3x^3(x - 2) = 3x^4 - 6x^3.
- We subtract the product from the dividend, which gives us (3x^4 - 4x^3 - 3x - 1) - (3x^4 - 6x^3) = 2x^3 - 3x.
- We bring down the next term of the dividend, which is -1, and repeat steps 2 to 4. We divide 2x^3 by x, which gives us a quotient of 2x^2. We write the quotient above the dividend and multiply it by the divisor, which gives us 2x^2(x - 2) = 2x^3 - 4x^2. We subtract the product from the dividend, which gives us (2x^3 - 3x) - (2x^3 - 4x^2) = 4