Divide the polynomials 9x^4+5x-3

Question

by 3x^2+1
to determine the quotient and remainder.

A
The quotient is 3x2+13x^2+13x
2
+1. The remainder is 5x−25x-25x−2.

B
The quotient is 3x2−13x^2-13x
2
−1. The remainder is 5x−45x-45x−4.

C
The quotient is 3x2+13x^2+13x
2
+1. The remainder is 5x+25x+25x+2.

D
The quotient is 3x2−13x^2-13x
2
−1. The remainder is 5x−25x-25x−2.

Answer 1

To divide the polynomial \( 9x^4 + 5x - 3 \) by \( 3x^2 + 1 \), we can use polynomial long division.

1. Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{9x^4}{3x^2} = 3x^2 \] This is the first term of the quotient.

2. Multiply the entire divisor by this term: \[ 3x^2 \cdot (3x^2 + 1) = 9x^4 + 3x^2 \]

3. Subtract this from the original polynomial: \[ (9x^4 + 5x - 3) - (9x^4 + 3x^2) = 5x - 3 - 3x^2 = -3x^2 + 5x - 3 \]

4. Repeat the process with the new polynomial \(-3x^2 + 5x - 3\): Divide the leading term of the new polynomial by the leading term of the divisor: \[ \frac{-3x^2}{3x^2} = -1 \] This becomes the second term of the quotient.

5. Multiply the entire divisor by this term: \[ -1 \cdot (3x^2 + 1) = -3x^2 - 1 \]

6. Subtract this from the new polynomial: \[ (-3x^2 + 5x - 3) - (-3x^2 - 1) = 5x - 3 + 1 = 5x - 2 \]

Now, \( 5x - 2 \) is our new polynomial, which has a degree lower than the divisor \( 3x^2 + 1 \). Thus we can stop here.

The final result of the division of \( 9x^4 + 5x - 3 \) by \( 3x^2 + 1 \) is:

• Quotient: \( 3x^2 - 1 \)
• Remainder: \( 5x - 2 \)

So the right answer is:

D: The quotient is \( 3x^2 - 1 \). The remainder is \( 5x - 2 \).