To use the long division method to divide the polynomial \( x^3 + 9x^2 + 21x + 9 \) by the polynomial \( x + 3 \), we can follow these steps:
Step 1: Set up the long division
We write it in the long division format:
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x + 3 | x^3 + 9x^2 + 21x + 9
Step 2: Divide
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First term: Divide the first term of the dividend \( x^3 \) by the first term of the divisor \( x \): \[ \frac{x^3}{x} = x^2 \] Write \( x^2 \) above the division line.
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Multiply: Multiply \( x^2 \) by \( x + 3 \): \[ x^2(x + 3) = x^3 + 3x^2 \] Write this product beneath the dividend.
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Subtract: Subtract this result from the original polynomial: \[ (x^3 + 9x^2) - (x^3 + 3x^2) = 6x^2 \] Bring down the next term \( + 21x \): \[ 6x^2 + 21x \]
Step 3: Repeat the process
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Next term: Divide the first term \( 6x^2 \) by \( x \): \[ \frac{6x^2}{x} = 6x \] Write \( +6x \) above the division line.
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Multiply: Multiply \( 6x \) by \( x + 3 \): \[ 6x(x + 3) = 6x^2 + 18x \] Write this product beneath \( 6x^2 + 21x \) and subtract: \[ (6x^2 + 21x) - (6x^2 + 18x) = 3x \] Bring down the next term \( + 9 \): \[ 3x + 9 \]
Step 4: One more time
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Next term: Divide \( 3x \) by \( x \): \[ \frac{3x}{x} = 3 \] Write \( +3 \) above the division line.
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Multiply: Multiply \( 3 \) by \( x + 3 \): \[ 3(x + 3) = 3x + 9 \] Subtract: \[ (3x + 9) - (3x + 9) = 0 \]
Final Result
Since the remainder is zero, the final result of dividing \( x^3 + 9x^2 + 21x + 9 \) by \( x + 3 \) is: \[ \boxed{x^2 + 6x + 3} \]
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