Questions: Find the quotient of: (x^3+4x^2-19x+14) ÷ (x+7)

Find the quotient of: (x^3+4x^2-19x+14) ÷ (x+7)
Transcript text: Find the quotient of: \[ \left(x^{3}+4 x^{2}-19 x+14\right) \div(x+7) \]
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Solution

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Solution Steps

Step 1: Polynomial Long Division

To find the quotient of the polynomial \(x^{3} + 4x^{2} - 19x + 14\) divided by \(x + 7\), we perform polynomial long division.

  1. Divide \(x^{3}\) by \(x\), resulting in \(x^{2}\).
  2. Multiply \(x^{2}\) by \(x + 7\) to get \(x^{3} + 7x^{2}\).
  3. Subtract this from the original polynomial: \[ (x^{3} + 4x^{2} - 19x + 14) - (x^{3} + 7x^{2}) = -3x^{2} - 19x + 14 \]
Step 2: Continue Division

Next, we continue the division with the new polynomial \(-3x^{2} - 19x + 14\).

  1. Divide \(-3x^{2}\) by \(x\), resulting in \(-3x\).
  2. Multiply \(-3x\) by \(x + 7\) to get \(-3x^{2} - 21x\).
  3. Subtract this from the current polynomial: \[ (-3x^{2} - 19x + 14) - (-3x^{2} - 21x) = 2x + 14 \]
Step 3: Final Division Step

Now, we divide \(2x + 14\) by \(x + 7\).

  1. Divide \(2x\) by \(x\), resulting in \(2\).
  2. Multiply \(2\) by \(x + 7\) to get \(2x + 14\).
  3. Subtract this from the current polynomial: \[ (2x + 14) - (2x + 14) = 0 \]

Final Answer

The quotient of the division is \(x^{2} - 3x + 2\) and the remainder is \(0\). Therefore, we can express the result of the division as: \[ \frac{x^{3} + 4x^{2} - 19x + 14}{x + 7} = x^{2} - 3x + 2 \]

Thus, the final answer is: \[ \boxed{x^{2} - 3x + 2} \]

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