QuestionAugust 27, 2025

What is the quotient (6x^4+15x^3-2x^2+10x-4)div (3x^2+2) ? 2x^2-5x+2 2x^2+5x-2 2x^2-5x-2 2x^2+5x+2

What is the quotient (6x^4+15x^3-2x^2+10x-4)div (3x^2+2) ? 2x^2-5x+2 2x^2+5x-2 2x^2-5x-2 2x^2+5x+2
What is the quotient (6x^4+15x^3-2x^2+10x-4)div (3x^2+2) ?
2x^2-5x+2
2x^2+5x-2
2x^2-5x-2
2x^2+5x+2

Solution
4.2(217 votes)

Answer

2x^{2}+5x-2 Explanation 1. Perform Polynomial Long Division Divide the first term of the dividend 6x^4 by the first term of the divisor 3x^2, which gives 2x^2. Multiply the entire divisor 3x^2 + 2 by 2x^2 and subtract from the original polynomial. 2. Subtract and Bring Down Next Term After subtraction, the new polynomial is 15x^3 - 2x^2 + 10x - 4. Repeat the division process with 15x^3 divided by 3x^2, giving 5x. Multiply and subtract again. 3. Continue Division Process The next polynomial becomes -2x^2 + 10x - 4. Divide -2x^2 by 3x^2, resulting in -\frac{2}{3}. Multiply and subtract to find the remainder. 4. Verify Remainder Ensure that the remainder is less than the degree of the divisor, confirming the quotient is correct.

Explanation

1. Perform Polynomial Long Division<br /> Divide the first term of the dividend $6x^4$ by the first term of the divisor $3x^2$, which gives $2x^2$. Multiply the entire divisor $3x^2 + 2$ by $2x^2$ and subtract from the original polynomial.<br />2. Subtract and Bring Down Next Term<br /> After subtraction, the new polynomial is $15x^3 - 2x^2 + 10x - 4$. Repeat the division process with $15x^3$ divided by $3x^2$, giving $5x$. Multiply and subtract again.<br />3. Continue Division Process<br /> The next polynomial becomes $-2x^2 + 10x - 4$. Divide $-2x^2$ by $3x^2$, resulting in $-\frac{2}{3}$. Multiply and subtract to find the remainder.<br />4. Verify Remainder<br /> Ensure that the remainder is less than the degree of the divisor, confirming the quotient is correct.
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