How do you divide the polynomial x^4 + 9x^2 – 9 by x^2 + 3x, and what are the quotient and the remainder?

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Dividing Polynomials: A Step-by-Step Guide

To divide the polynomial x4 + 9x2 – 9 by x2 + 3x, we can use polynomial long division. Here’s how to perform the division:

Step 1: Set Up the Division

Write the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by) in a long division format:

       ______________________
    x2 + 3x | x4 + 0x3 + 9x2 + 0x - 9

Step 2: Divide the Leading Terms

The leading term of the dividend is x4 and the leading term of the divisor is x2. Divide these to find the first term of the quotient:

x4 ÷ x2 = x2

Step 3: Multiply and Subtract

Multiply the entire divisor by x2: 

  x2 * (x2 + 3x) = x4 + 3x3

Now, subtract this from the original polynomial:

       x4 + 0x3 + 9x2 + 0x - 9
      - (x4 + 3x3)
       ______________________
                 -3x3 + 9x2 + 0x - 9

Step 4: Repeat the Process

The next leading term is -3x3. Divide this by x2:

-3x3 ÷ x2 = -3x

Now multiply the entire divisor again:

  -3x * (x2 + 3x) = -3x3 - 9x2

Subtract this from the current polynomial:

        -3x3 + 9x2 + 0x - 9
       - (-3x3 - 9x2) 
        ______________________
                 18x2 + 0x - 9

Step 5: Final Division

Now, we have 18x2 as the next leading term:

18x2 ÷ x2 = 18

Multiply the divisor:

  18 * (x2 + 3x) = 18x2 + 54x

Subtract this from the polynomial:

        18x2 + 0x - 9
       - (18x2 + 54x)
        ______________________
                   -54x - 9

Step 6: Conclusion

At this point, the degree of the remainder (-54x – 9) is less than the degree of the divisor, which means we cannot divide further.

Final Result

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

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