Question

Divide the polynomials 9x^4+5x-3
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 \( 5x - 2 \).