Using long division, divide x3−3x2−13x+15
by ( x+3
)
To divide \(x^3 - 3x^2 - 13x + 15\) by \(x + 3\), we perform the following long division:
x^2 - 6x + 13
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x + 3 | x^3 - 3x^2 - 13x + 15
- (x^3 + 3x^2)
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-6x + 15
- (-6x - 18)
______________
33
Therefore, the quotient is \(x^2 - 6x + 13\) and the remainder is 33.
To divide the polynomial x^3 - 3x^2 - 13x + 15 by x + 3 using long division, follow these steps:
Step 1: Write the dividend and divisor in the long division format:
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x + 3 | x^3 - 3x^2 - 13x + 15
Step 2: Divide the first term of the dividend (x^3) by the first term of the divisor (x).
x^3 ÷ x = x^2
Step 3: Multiply the result obtained in Step 2 (x^2) by the divisor (x + 3) and write the result below the dividend.
x^2 * (x + 3) = x^3 + 3x^2
Step 4: Subtract the result obtained in Step 3 from the dividend.
(x^3 - 3x^2 - 13x + 15) - (x^3 + 3x^2) = -6x^2 - 13x + 15
Step 5: Bring down the next term from the dividend (-6x^2) and rewrite the expression.
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x + 3 | x^3 - 3x^2 - 13x + 15 -6x^2 - 13x +15
Step 6: Divide the first term of the new expression (-6x^2) by the first term of the divisor (x).
-6x^2 ÷ x = -6x
Step 7: Multiply the result obtained in Step 6 (-6x) by the divisor (x + 3) and write the result below the new expression.
-6x * (x + 3) = -6x^2 - 18x
Step 8: Subtract the result obtained in Step 7 from the new expression.
(-6x^2 - 13x + 15) - (-6x^2 - 18x) = 5x + 15
Step 9: Bring down the next term from the original dividend (5x) and rewrite the expression.
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x + 3 | x^3 - 3x^2 - 13x + 15 - 6x^2 - 13x + 15 5x + 15
Step 10: Divide the first term of the new expression (5x) by the first term of the divisor (x).
5x ÷ x = 5
Step 11: Multiply the result obtained in Step 10 (5) by the divisor (x + 3) and write the result below the new expression.
5 * (x + 3) = 5x + 15
Step 12: Subtract the result obtained in Step 11 from the new expression.
(5x + 15) - (5x + 15) = 0
Step 13: There is no remainder, so the long division is complete.
The quotient is x^2 - 6x + 5.
To divide the polynomial x^3 - 3x^2 - 13x + 15 by x + 3 using long division, follow these steps:
Step 1: Arrange the terms of the dividend in descending order. The dividend is x^3 - 3x^2 - 13x + 15.
Step 2: Divide the first term of the dividend by the first term of the divisor. Divide x^3 by x, which gives x^2.
Step 3: Multiply the divisor (x + 3) by the result from step 2 (x^2). The product is (x^2)(x + 3) = x^3 + 3x^2.
Step 4: Subtract the result from step 3 from the original dividend. (x^3 - 3x^2 - 13x + 15) - (x^3 + 3x^2) = -6x^2 - 13x + 15.
Step 5: Bring down the next term from the original dividend, which is -6x^2. Now the new dividend is -6x^2 - 13x + 15.
Step 6: Divide the first term of the new dividend (-6x^2) by the first term of the divisor (x). The result is -6x.
Step 7: Multiply the divisor (x + 3) by the result from step 6 (-6x). The product is (-6x)(x + 3) = -6x^2 - 18x.
Step 8: Subtract the result from step 7 from the updated dividend. (-6x^2 - 13x + 15) - (-6x^2 - 18x) = 5x + 15.
Step 9: Bring down the next term from the updated dividend, which is 5x. Now the new dividend is 5x + 15.
Step 10: Divide the first term of the new dividend (5x) by the first term of the divisor (x). The result is 5.
Step 11: Multiply the divisor (x + 3) by the result from step 10 (5). The product is (5)(x + 3) = 5x + 15.
Step 12: Subtract the result from step 11 from the updated dividend. (5x + 15) - (5x + 15) = 0.
Step 13: There are no more terms to bring down. The division is complete.
The final result of dividing x^3 - 3x^2 - 13x + 15 by x + 3 is x^2 - 6x + 5.