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Answer :
Answer:
the answer is 6x^3 + 27x - 19x^2 - 5/x -5
Step-by-step explanation:
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Rewritten by : Barada
By dividing [tex]6x^3+27x-19x^2-15[/tex] by [tex]3x-5[/tex] we get [tex]\[2x^2 - 3x + 4 + \frac{5}{3x - 5}\][/tex]
To divide the polynomial [tex]\( 6x^3 + 27x - 19x^2 - 15 \) by \( 3x - 5 \),[/tex] we will use polynomial long division. Let's go through the steps:
1. Arrange the polynomials in standard form:
Dividend:,,[tex]\( 6x^3 - 19x^2 + 27x - 15 \)[/tex]
Divisor: 3x - 5
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\( \frac{6x^3}{3x} = 2x^2 \)[/tex]
3. Multiply the entire divisor by the result and subtract from the dividend:
[tex]\( (3x - 5) \times 2x^2 = 6x^3 - 10x^2 \)[/tex]
Subtract: [tex]\( (6x^3 - 19x^2 + 27x - 15) - (6x^3 - 10x^2) = -9x^2 + 27x - 15 \)[/tex]
4. Repeat the process with the new polynomial:
Divide the first term of the new polynomial by the first term of the divisor: [tex]\( (6x^3 - 19x^2 + 27x - 15) - (6x^3 - 10x^2) = -9x^2 + 27x - 15 \)[/tex]
Multiply the entire divisor by this result:[tex]\( (3x - 5) \times -3x = -9x^2 + 15x \)[/tex]
Subtract: [tex]\( (-9x^2 + 27x - 15) - (-9x^2 + 15x) = 12x - 15 \)[/tex]
5. Repeat the process again:
Divide the first term of the new polynomial by the first term of the divisor: [tex]\( \frac{12x}{3x} = 4 \)[/tex]
Multiply the entire divisor by this result:[tex]\( (3x - 5) \times 4 = 12x - 20 \)[/tex]
Subtract: (12x - 15) - (12x - 20) = 5
So, the quotient is [tex]\( 2x^2 - 3x + 4 \)[/tex] and the remainder is 5
Therefore, we can write the result as:
[tex]\[\frac{6x^3 + 27x - 19x^2 - 15}{3x - 5} = 2x^2 - 3x + 4 + \frac{5}{3x - 5}\][/tex]
So, the final answer is:
[tex]\[2x^2 - 3x + 4 + \frac{5}{3x - 5}\][/tex]
