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vIf we divide the polynomial x^(4) + 4x^(3) + 2x^(2) + x + 4 by x^(2) + 3x, what will be the remainder?

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Answer:
To divide the polynomial x^(4) + 4x^(3) + 2x^(2) + x + 4 by x^(2) + 3x, we can use polynomial long division

Explanation:
Step 1: Divide the leading term of the dividend (x^(4)) by the leading term of the divisor (x^(2)). The result is x^(2).

Step 2: Multiply the entire divisor (x^(2) + 3x) by the result from Step 1 (x^(2)). The result is x^(4) + 3x^(3).

Step 3: Subtract the result from Step 2 from the dividend (x^(4) + 4x^(3) + 2x^(2) + x + 4). The subtraction gives us x^(4) + 4x^(3) + 2x^(2) + x + 4 - (x^(4) + 3x^(3)), which simplifies to x^(3) + 2x^(2) + x + 4.

Step 4: Repeat Steps 1-3 with the new simplified dividend (x^(3) + 2x^(2) + x + 4).

Step 5: Divide the leading term of the new simplified dividend (x^(3)) by the leading term of the divisor (x^(2)). The result is x.

Step 6: Multiply the entire divisor (x^(2) + 3x) by the result from Step 5 (x). The result is x^(3) + 3x^(2).

Step 7: Subtract the result from Step 6 from the new simplified dividend (x^(3) + 2x^(2) + x + 4). The subtraction gives us x^(3) + 2x^(2) + x + 4 - (x^(3) + 3x^(2)), which simplifies to -x^(2) + x + 4.

Since we can no longer divide further (the degree of the new simplified dividend is less than the degree of the divisor), the remainder is -x^(2) + x + 4

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