The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x). Complete the division process and find the remainder of this polynomial division.
The polynomial division of (3x^4 + 9x^2 - 13) ÷ (x^2 - 5x) can be completed as follows:
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x^2 - 5x | 3x^4 + 9x^2 + 0x - 13
- (3x^4 - 15x^3)
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24x^3 + 0x^2
Now, we continue with the division:
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x^2 - 5x | 3x^4 + 9x^2 + 0x - 13
- (3x^4 - 15x^3)
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24x^3 + 0x^2 - 13
- (24x^3 - 120x^2)
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120x^2 - 13
Since no further division is possible, we can conclude that the remainder of the polynomial division is 120x^2 - 13.
To complete the polynomial division, we need to continue dividing until there are no more terms to bring down or the degree of the remainder is less than the degree of the divisor.
Let's continue with the division:
3x^2
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x^2 - 5x | 3x^4 + 0x^3 + 9x^2 + 0x - 13
- (3x^4 - 15x^3)
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15x^3 + 9x^2
Now, bring down the next term:
15x^3 + 9x^2
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x^2 - 5x | 3x^4 + 0x^3 + 9x^2 + 0x - 13
- (3x^4 - 15x^3)
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15x^3 + 9x^2 + 0x
- (15x^3 - 75x^2)
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84x^2 + 0x
Since the degree of the remainder, 84x^2, is less than the degree of the divisor, x^2 - 5x, we can stop here.
Therefore, the remainder of the polynomial division (3x^4 + 9x^2 - 13) ÷ (x^2 - 5x) is 84x^2.