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.
(1 point)
Responses
The remainder is 420x−13.
The remainder is 420 x minus 13 .
The remainder is 2,087.
The remainder is 2,087.
The remainder is 407.
The remainder is 407.
The remainder is 407x.
The remainder is -13.
To complete the polynomial division of (3x^4+9x^2-13)÷(x^2-5x), we continue the division process:
3x^2 + 15x
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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
- (24x^3 - 120x^2)
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120x^2 + 0x
- (120x^2 - 600x)
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600x - 13
- (600x - 3,000)
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- 2,987
Therefore, the remainder of the polynomial division is -2,987.
To complete the polynomial division, we need to follow these steps:
1. Start by dividing the first term of the dividend (3x^4) by the first term of the divisor (x^2). This gives us 3x^2 as the first term of the quotient.
2. Multiply the entire divisor (x^2 - 5x) by the first term of the quotient (3x^2) and subtract it from the dividend (3x^4 + 9x^2 - 13). This gives us:
(3x^4 + 9x^2 - 13) - (3x^2)(x^2 - 5x) = 3x^4 + 9x^2 - 13 - (3x^4 - 15x^3) = 9x^2 + 15x^3 - 13.
3. Repeat steps 1 and 2 with the new dividend (9x^2 + 15x^3 - 13). Divide the first term of the new dividend (15x^3) by the first term of the divisor (x^2). This gives us 15x as the second term of the quotient.
4. Multiply the entire divisor (x^2 - 5x) by the second term of the quotient (15x) and subtract it from the new dividend (9x^2 + 15x^3 - 13). This gives us:
(9x^2 + 15x^3 - 13) - (15x)(x^2 - 5x) = 9x^2 + 15x^3 - 13 - (15x^3 - 75x^2) = -66x^2 - 13.
5. Repeat steps 1 and 2 with the new dividend (-66x^2 - 13). Divide the first term of the new dividend (-66x^2) by the first term of the divisor (x^2). This gives us -66 as the third term of the quotient.
6. Multiply the entire divisor (x^2 - 5x) by the third term of the quotient (-66) and subtract it from the new dividend (-66x^2 - 13). This gives us:
(-66x^2 - 13) - (-66)(x^2 - 5x) = -66x^2 - 13 + (66x^2 - 330x) = -330x - 13.
After repeating these steps until we cannot divide any further, we find that the remainder of the polynomial division is -330x - 13.