Question

What are the quotient and remainder when 12x² + 11x + 2 is divided by 3x²?

1) Quotient: 4x + 1, Remainder: 2
2) Quotient: 4x + 1, Remainder: 0
3) Quotient: 3x + 2, Remainder: 1
4) Quotient: 3x + 2, Remainder: 0

Answer 1

The division of 12x²+ 11x + 2 by 3x² results in a quotient of 4x + 1 and a remainder of 2. Thus the correct option is 1.

Step-by-step explanation:

To find the quotient and remainder, we use polynomial long division. Starting with the leading terms, 12x² divided by 3x² gives 4x, which is the leading term of the quotient. Multiplying 3x² by 4x gives 12x³, and subtracting this from the numerator leaves 11x + 2.

Now, we repeat the process with 11x + 2, dividing it by 3x². The next term in the quotient is +1, and multiplying 3x² by 4x + 1 gives 12x² + 3x. Subtracting this from 11x + 2 gives a remainder of 2. Therefore, the complete quotient is 4x + 1 with a remainder of 2.

Polynomial long division is a systematic method used to divide polynomials and obtain both the quotient and remainder. It involves dividing the leading terms, subtracting the product from the numerator, and repeating the process until the degree of the remainder is less than the degree of the divisor. In this case, the division yields a quotient of 4x + 1 and a remainder of 2.