Questions: (6x^2+11x-10) ÷ (3x-2)

(6x^2+11x-10) ÷ (3x-2)
Transcript text: 5. $\left(6 x^{2}+11 x-10\right) \div(3 x-2)$
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Solution

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Solution Steps

To solve the polynomial division problem, we can use polynomial long division or synthetic division. In this case, we will use polynomial long division to divide \(6x^2 + 11x - 10\) by \(3x - 2\).

Step 1: Define the Polynomials

We start with the polynomial \(6x^2 + 11x - 10\) as the numerator and \(3x - 2\) as the denominator.

Step 2: Perform Polynomial Division

Using polynomial long division, we divide \(6x^2 + 11x - 10\) by \(3x - 2\).

  1. Divide the leading term of the numerator \(6x^2\) by the leading term of the denominator \(3x\) to get \(2x\).

  2. Multiply \(2x\) by the entire denominator \(3x - 2\) to get \(6x^2 - 4x\).

  3. Subtract this result from the original numerator: \[ (6x^2 + 11x - 10) - (6x^2 - 4x) = 15x - 10 \]

  4. Now, divide the leading term \(15x\) by \(3x\) to get \(5\).

  5. Multiply \(5\) by the entire denominator \(3x - 2\) to get \(15x - 10\).

  6. Subtract this from the current remainder: \[ (15x - 10) - (15x - 10) = 0 \]

Step 3: Write the Result

The quotient from the division is \(2x + 5\) and the remainder is \(0\). Therefore, we can express the result of the division as: \[ \frac{6x^2 + 11x - 10}{3x - 2} = 2x + 5 \]

Final Answer

\(\boxed{2x + 5}\)

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