(3x4+9x2−13)÷(x2−5x). Complete the division process and find the remainder of this polynomial division.
To find the remainder of this polynomial division, we need to divide the given polynomial, which is 3x^4 + 9x^2 - 13, by the divisor, which is x^2 - 5x.
The first step is to rewrite the polynomial division problem in the correct form:
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x^2 - 5x | 3x^4 + 9x^2 + 0x - 13
To start dividing, we will compare the first term of the dividend (3x^4) with the first term of the divisor (x^2).
The quotient will be (3x^4/x^2), which is 3x^2. We then multiply this quotient by the divisor, which gives us the product (3x^2 * (x^2 - 5x)), which is 3x^4 - 15x^3.
Next, we subtract this product from the dividend:
3x^4 + 9x^2 - 13 - (3x^4 - 15x^3)
Simplifying, we get:
3x^4 + 9x^2 - 13 - 3x^4 + 15x^3
Combining like terms, we have:
15x^3 + 9x^2 - 13
We now repeat the process, comparing the first term of the new dividend (15x^3) with the first term of the divisor (x^2).
The new quotient will be (15x^3/x^2), which is 15x. We then multiply this quotient by the divisor, which gives us the product (15x * (x^2 - 5x)), which is 15x^3 - 75x^2.
Subtracting this product from the new dividend, we have:
15x^3 + 9x^2 - 13 - (15x^3 - 75x^2)
Simplifying, we get:
15x^3 + 9x^2 - 13 - 15x^3 + 75x^2
Combining like terms once again, we have:
84x^2 - 13.
Since there are no more terms with a degree higher than the divisor, we stop the division process.
Therefore, the remainder of this polynomial division is 84x^2 - 13.
To perform polynomial division, we need to divide the numerator polynomial (3x^4 + 9x^2 - 13) by the denominator polynomial (x^2 - 5x).
First, let's write the division equation in the proper format:
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
Now, let's perform the division step-by-step:
1. Divide the leading term of the numerator by the leading term of the denominator: (3x^4 / x^2) = 3x^2.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
2. Multiply the quotient obtained in the previous step by the entire denominator: 3x^2 * (x^2 - 5x) = 3x^4 - 15x^3.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3
3. Subtract the product obtained in the previous step from the numerator: (9x^2 - 24x^3) - (3x^4 - 15x^3) = -3x^4 + 39x^3 + 9x^2.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3 + 9x^2
4. Repeat the process, dividing the leading term of the new numerator (-3x^4 + 39x^3 + 9x^2) by the leading term of the denominator (x^2): (-3x^4 / x^2) = -3x^2.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3 + 9x^2
- (-3x^4 + 15x^3)
5. Multiply the new quotient (-3x^2) by the entire denominator: -3x^2 * (x^2 - 5x) = -3x^4 + 15x^3.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3 + 9x^2
- (-3x^4 + 15x^3)
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-6x^3 + 9x^2
6. Subtract the new product from the previous operation from the numerator: (39x^3 + 9x^2) - (-6x^3 + 9x^2) = 45x^3.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3 + 9x^2
- (-3x^4 + 15x^3)
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-6x^3 + 9x^2
- ( -6x^3 + 30x^2)
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-21x^2
7. Repeat the process, dividing the leading term of the new numerator (45x^3) by the leading term of the denominator (x^2): (45x^3 / x^2) = 45x.
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(x^2 - 5x) | (3x^4 + 9x^2 - 13)
- (3x^4 - 15x^3)
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24x^3 + 9x^2
- (-3x^4 + 15x^3)
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-6x^3 + 9x^2
- ( -6x^3 + 30x^2)
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-21x^2
- ( -21x^2 + 105x)
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-105x - 13
Since there are no more terms in the numerator with a degree higher than the denominator, we can stop here.
Therefore, the remainder of the polynomial division is -105x - 13.