Question

Divide using long division. State the​ quotient, q(x), and the​ remainder, r(x).
StartFraction 16 x Superscript 4 Baseline plus 6 x cubed plus 14 x squared Over 2 x squared plus 3 EndFraction
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Part 1
StartFraction 16 x Superscript 4 Baseline plus 6 x cubed plus 14 x squared Over 2 x squared plus 3 EndFraction
equals
  
enter your response hereplusStartFraction nothing Over 2 x squared plus 3 EndFraction
  
​(Simplify your answers. Do not factor. Use integers or fractions for any numbers in the​ expressions.)

Answer 1

To perform the long division of the polynomial \( \frac{16x^4 + 6x^3 + 14x^2}{2x^2 + 3} \), we proceed by dividing step by step.

1. **Divide the leading term of the numerator by the leading term of the denominator:**
\[
\frac{16x^4}{2x^2} = 8x^2
\]
This is the first term of our quotient \( q(x) \).

2. **Multiply the entire denominator by this term:**
\[
(2x^2 + 3)(8x^2) = 16x^4 + 24x^2
\]

3. **Subtract this result from the original polynomial:**
\[
(16x^4 + 6x^3 + 14x^2) - (16x^4 + 24x^2) = 6x^3 + 14x^2 - 24x^2 = 6x^3 - 10x^2
\]

4. **Now repeat the process for the new polynomial \( 6x^3 - 10x^2 \):**
- Divide the leading term:
\[
\frac{6x^3}{2x^2} = 3x
\]
This is the next term in the quotient, so now we have \( q(x) = 8x^2 + 3x \).

5. **Multiply the entire denominator by \( 3x \):**
\[
(2x^2 + 3)(3x) = 6x^3 + 9x
\]

6. **Subtract this from the result of the previous step:**
\[
(6x^3 - 10x^2) - (6x^3 + 9x) = -10x^2 - 9x
\]

7. **Now for the polynomial \( -10x^2 - 9x \):**
- Divide the leading term:
\[
\frac{-10x^2}{2x^2} = -5
\]
This is the next term in the quotient, so now we have \( q(x) = 8x^2 + 3x - 5 \).

8. **Multiply the entire denominator by \( -5 \):**
\[
(2x^2 + 3)(-5) = -10x^2 - 15
\]

9. **Subtract this from the result of the previous step:**
\[
(-10x^2 - 9x) - (-10x^2 - 15) = -9x + 15
\]
This is our remainder \( r(x) \).

10. **Putting it all together:**
The final result of the division is:
\[
\frac{16x^4 + 6x^3 + 14x^2}{2x^2 + 3} = 8x^2 + 3x - 5 + \frac{-9x + 15}{2x^2 + 3}
\]

Thus, we conclude:
- **Quotient:** \( q(x) = 8x^2 + 3x - 5 \)
- **Remainder:** \( r(x) = -9x + 15 \)

This gives us:
\[
\frac{16x^4 + 6x^3 + 14x^2}{2x^2 + 3} = 8x^2 + 3x - 5 + \frac{-9x + 15}{2x^2 + 3}
\]