Question

what is the quotient of 2x^3 + 3x^2 + 5x - 4 divided by x^2 + x +1?

Answer 1

To find the quotient of \( 2x^3 + 3x^2 + 5x - 4 \) divided by \( x^2 + x + 1 \), we will perform polynomial long division.

1. **Division Step**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{2x^3}{x^2} = 2x
\]

2. **Multiply**: Multiply the entire divisor \( x^2 + x + 1 \) by \( 2x \):
\[
2x(x^2 + x + 1) = 2x^3 + 2x^2 + 2x
\]

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

4. **Next Division Step**: Now we repeat the process with the new polynomial \( x^2 + 3x - 4 \):
\[
\frac{x^2}{x^2} = 1
\]

5. **Multiply**: Multiply the divisor by \( 1 \):
\[
1(x^2 + x + 1) = x^2 + x + 1
\]

6. **Subtract**: Subtract this from \( x^2 + 3x - 4 \):
\[
(x^2 + 3x - 4) - (x^2 + x + 1) = (3x - x) + (-4 - 1) = 2x - 5
\]

7. **Final Output**: The degree of the remainder \( 2x - 5 \) is less than the degree of the divisor \( x^2 + x + 1 \). Thus, we stop here.

The final result of the long division is:
\[
\text{Quotient: } 2x + 1 \quad \text{Remainder: } 2x - 5
\]

In summary:
\[
\frac{2x^3 + 3x^2 + 5x - 4}{x^2 + x + 1} = 2x + 1 + \frac{2x - 5}{x^2 + x + 1}
\]

So the quotient is \( 2x + 1 \).