Dividing Polynomial x^4 + 9x^2 – 9 by x^2 + 3x: What are the Quotient and the Remainder?

To divide the polynomial x⁴ + 9x² – 9 by x² + 3x, we can use polynomial long division.

Firstly, we set it up similar to numeric long division:

• Dividend: x⁴ + 0x³ + 9x² + 0x – 9
• Divisor: x² + 3x

Next, we divide the leading term of the dividend by the leading term of the divisor:

• x⁴ ÷ x² = x²

Now, we multiply the entire divisor by x²:

• x² * (x² + 3x) = x⁴ + 3x³

Next, we subtract this result from the original polynomial:

• (x⁴ + 9x² – 9) – (x⁴ + 3x³) = -3x³ + 9x² – 9

Now we repeat the process with the new polynomial -3x³ + 9x² – 9.

Divide the leading term again:

• -3x³ ÷ x² = -3x

Multiply the divisor by -3x:

• -3x * (x² + 3x) = -3x³ – 9x²

Subtract once more:

• (-3x³ + 9x² – 9) – (-3x³ – 9x²) = 18x² – 9

Now, we divide the leading term again:

• 18x² ÷ x² = 18

Multiply the divisor by 18:

• 18 * (x² + 3x) = 18x² + 54x

Subtract again:

• (18x² – 9) – (18x² + 54x) = -54x – 9

At this point, we cannot divide anymore since the degree of the remaining polynomial -54x – 9 is less than the degree of the divisor. Thus, we identify our results:

The final quotient is x² – 3x + 18 and the remainder is -54x – 9.

To summarize:

• Quotient: x² – 3x + 18
• Remainder: -54x – 9