To simplify the expression \(\frac{x^3 + 5x^2 - 20}{x + 3}\), we can use polynomial long division.
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Divide the first term of the numerator by the first term of the denominator:
- \(x^3 \div x = x^2\)
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Multiply the entire divisor by this result:
- \((x + 3)x^2 = x^3 + 3x^2\)
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Subtract this from the original polynomial:
- \((x^3 + 5x^2 - 20) - (x^3 + 3x^2) = 2x^2 - 20\)
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Repeat with the new polynomial:
- Divide \(2x^2\) by \(x\) to get \(2x\).
- Multiply the entire divisor by \(2x\): \((x + 3)(2x) = 2x^2 + 6x\).
- Subtract: \((2x^2 - 20) - (2x^2 + 6x) = -6x - 20\).
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Repeat again:
- Divide \(-6x\) by \(x\) to get \(-6\).
- Multiply the entire divisor by \(-6\): \((x + 3)(-6) = -6x - 18\).
- Subtract: \((-6x - 20) - (-6x - 18) = -2\).
Now we've completed the polynomial long division. We have: \[ x^3 + 5x^2 - 20 = (x + 3)(x^2 + 2x - 6) - 2 \] This means our simplified expression is: \[ \frac{x^3 + 5x^2 - 20}{x + 3} = x^2 + 2x - 6 - \frac{2}{x + 3} \] However, since we're just simplifying to the polynomial part without the remainder: \[ = x^2 + 2x - 6 \]
Now, to fill in the blanks based on this:
- We can write \(x^2 + 2x - 6\) in the form of \(a + bx + cx^2\):
- \(Blank 1 = -6\)
- \(Blank 5 = 2\)
- \(Blank 7 = 1\) (the coefficient of \(x^2\) is 1)
Thus the completed equation from the options provided is:
- Blank space 1: -6
- Blank space 5: 2
- Blank space 7: 1
The fourth space was not needed based on the simplification path provided, and thus is empty.
Final Answer:
- Blank space 1: -6
- Blank space 5: 2
- Blank space 7: 1
- Blank space 8: [Empty]
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