To perform the polynomial division \((x^3 + 5x^2 + 4x + 18) \div (x + 5)\), we will divide the polynomial \(x^3 + 5x^2 + 4x + 18\) by \(x + 5\) using long division.
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Divide the leading term:
- Divide the leading term \(x^3\) by the leading term \(x\): \[ x^3 \div x = x^2 \]
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Multiply and subtract:
- Multiply \(x^2\) by \((x + 5)\): \[ x^2(x + 5) = x^3 + 5x^2 \]
- Subtract this from the original polynomial: \[ (x^3 + 5x^2 + 4x + 18) - (x^3 + 5x^2) = 4x + 18 \]
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Repeat the process:
- Now, divide the new leading term \(4x\) by \(x\): \[ 4x \div x = 4 \]
- Multiply \(4\) by \((x + 5)\): \[ 4(x + 5) = 4x + 20 \]
- Subtract this from \(4x + 18\): \[ (4x + 18) - (4x + 20) = 18 - 20 = -2 \]
Now that we have completed the division, the quotient is: \[ x^2 + 4 \] And the remainder is: \[ -2 \]
- Express the remainder as a fraction: Therefore, we can express the final result of the polynomial division as: \[ x^2 + 4 - \frac{2}{x + 5} \]
Combining all parts, the final quotient with the remainder expressed as a fraction is: \[ \boxed{x^2 + 4 - \frac{2}{x + 5}} \]