Question

What is the remainder when (3x⁴ - 2x³ - x² + 2x - 9) ÷ (x²)?

1) 0
2) 5
3) 10
4) 15

Answer 1

The remainder of the division (3x⁴ - 2x³ - x² + 2x - 9) ÷ (x²) is found to be 2x - 9 after performing polynomial long division. The numerical remainder is -9, which does not match any of the given options.

Step-by-step explanation:

The remainder when dividing a polynomial by another polynomial is found by performing polynomial long division. In this case, we are dividing the polynomial (3x⁴ - 2x³ - x² + 2x - 9) by (x²). Here's how we approach it:

• Divide the first term of the numerator by the first term of the denominator. This will be the first term of the quotient.
• Then multiply the entire denominator by the new term found and subtract this from the numerator.
• Repeat this process until the degree of the remainder is less than the degree of the denominator, or no terms are left.

Applying this process to the given polynomial:

1. We first divide 3x⁴ by x² to get 3x² as the first term of the quotient.
2. 3x² times x² equals 3x⁴, which we subtract from the numerator leaving us with (-2x³ - x² + 2x - 9).
3. Next, we divide -2x³ by x² to get -2x as the new term of the quotient.
4. Multiplying -2x² and subtracting from what's left of the polynomial, we end up with -x² + 2x - 9.
5. Lastly, we divide -x² by x² to get a new term of -1.
6. Multiplying -1 with x² and subtracting, we're now left with a remainder of 2x - 9.

As the degree of the remainder is less than the degree of the denominator, long division is complete and the remainder is 2x - 9. Since the question asks for a numerical remainder, it means it's looking for the constant term of the remainder when the division has no x term left; hence, the answer is -9. None of the options provided matches this result, implying a possible mistake in the question options.