Question

What is the quotient of (24x4+18x3−9x2+3)÷(4x2−3x+1)

Answer 1

To perform the division of the polynomial \(24x^4 + 18x^3 - 9x^2 + 3\) by \(4x^2 - 3x + 1\), we'll use polynomial long division.

1. **Set up the division**: We write \(24x^4 + 18x^3 - 9x^2 + 0x + 3\) under the long division symbol and \(4x^2 - 3x + 1\) on the outside.

2. **Divide the leading terms**:
- Divide the leading term of the dividend \(24x^4\) by the leading term of the divisor \(4x^2\):
\[
\frac{24x^4}{4x^2} = 6x^2
\]

3. **Multiply and subtract**:
- Multiply \(6x^2\) by the entire divisor \(4x^2 - 3x + 1\):
\[
6x^2(4x^2 - 3x + 1) = 24x^4 - 18x^3 + 6x^2
\]
- Subtract this result from the original polynomial:
\[
(24x^4 + 18x^3 - 9x^2 + 3) - (24x^4 - 18x^3 + 6x^2) = 36x^3 - 15x^2 + 3
\]

4. **Repeat the process**:
- Now we take the new polynomial \(36x^3 - 15x^2 + 3\).
- Divide the leading term \(36x^3\) by the leading term \(4x^2\):
\[
\frac{36x^3}{4x^2} = 9x
\]
- Multiply \(9x\) by the divisor \(4x^2 - 3x + 1\):
\[
9x(4x^2 - 3x + 1) = 36x^3 - 27x^2 + 9x
\]
- Subtract this from \(36x^3 - 15x^2 + 3\):
\[
(36x^3 - 15x^2 + 3) - (36x^3 - 27x^2 + 9x) = 12x^2 - 9x + 3
\]

5. **Repeat again**:
- Take \(12x^2 - 9x + 3\).
- Divide \(12x^2\) by \(4x^2\):
\[
\frac{12x^2}{4x^2} = 3
\]
- Multiply \(3\) by \(4x^2 - 3x + 1\):
\[
3(4x^2 - 3x + 1) = 12x^2 - 9x + 3
\]
- Subtract:
\[
(12x^2 - 9x + 3) - (12x^2 - 9x + 3) = 0
\]

No remainder is left.

6. **Conclusion**:
The quotient is:
\[
6x^2 + 9x + 3
\]

Therefore, the result of dividing \(24x^4 + 18x^3 - 9x^2 + 3\) by \(4x^2 - 3x + 1\) is:
\[
6x^2 + 9x + 3
\]