Divide the polynomials 2x^3−x^2−13x+22 by x^2+x−5 to determine the quotient and remainder.(1 point)
1. The quotient is 2x-3. The remainder is -20x+7.
2. The quotient is 2x+1. The remainder is 7.
3. The quotient is 2x-3. The remainder is 7.
4. The quotient is 2x+1. The remainder is -4x+27.
The correct answer is 1. The quotient is 2x-3. The remainder is -20x+7.
To divide the polynomials 2x^3 - x^2 - 13x + 22 by x^2 + x - 5, follow these steps:
Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor.
In this case, divide 2x^3 by x^2. The result is 2x.
Step 2: Multiply the divisor (x^2 + x - 5) by the result from step 1 (2x) and subtract the product from the dividend (2x^3 - x^2 - 13x + 22).
(2x)(x^2 + x - 5) = 2x^3 + 2x^2 - 10x
(2x^3 - x^2 - 13x + 22) - (2x^3 + 2x^2 - 10x) = -3x^2 - 3x + 22
Step 3: Repeat steps 1 and 2 with the new polynomial (-3x^2 - 3x + 22).
Step 4: Divide the highest degree term of the new polynomial (-3x^2) by the highest degree term of the divisor (x^2). The result is -3x.
Step 5: Multiply the divisor (x^2 + x - 5) by the result from step 4 (-3x) and subtract the product from the new polynomial (-3x^2 - 3x + 22).
(-3x)(x^2 + x - 5) = -3x^3 - 3x^2 + 15x
(-3x^2 - 3x + 22) - (-3x^3 - 3x^2 + 15x) = 12x^2 - 18x + 22
Step 6: Repeat steps 4 and 5 with the new polynomial (12x^2 - 18x + 22).
Since the degree of the resulting polynomial, 12x^2 - 18x + 22, is less than the degree of the divisor (x^2 + x - 5), we stop here.
The quotient is the sum of the results obtained in steps 1 and 4: 2x - 3x = -x.
The remainder is the final polynomial obtained in step 6: 12x^2 - 18x + 22.
Therefore, the correct answer is:
The quotient is -x and the remainder is 12x^2 - 18x + 22.
To divide the polynomials 2x^3−x^2−13x+22 by x^2+x−5, you can use the polynomial long division method. Here are the steps to follow:
1. Set up the division with the divisor (x^2+x−5) on the left and the dividend (2x^3−x^2−13x+22) on the right. Write the terms in descending order of degree.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
2. Divide the highest-degree term of the dividend (2x^3) by the highest-degree term of the divisor (x^2). The result is the first term of the quotient.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
2x
3. Multiply the divisor (x^2+x−5) by the first term of the quotient (2x). Write the result below the dividend, aligning like-terms.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
4. Subtract the result from step 3 from the dividend. This gives you a new polynomial to continue dividing.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
___________________
- 13x + 22
5. Bring down the next term from the dividend (-13x).
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
___________________
- 13x + 22
- 13x
6. Divide the highest-degree term of the new polynomial (- 13x) by the highest-degree term of the divisor (x^2). The result is the next term of the quotient.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
___________________
- 13x + 22
- 13x
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9x + 22
7. Multiply the divisor (x^2+x−5) by the newly obtained term (9x). Write the result below the previous result, aligning like-terms.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
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- 13x + 22
- 13x
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9x + 22
+ 9x^2
- 45x
8. Subtract the result from step 7 from the polynomial obtained in step 6. This gives you the final remainder.
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x^2+x-5 | 2x^3 - x^2 - 13x + 22
- (2x^3 + 2x^2 - 10x)
___________________
- 13x + 22
- 13x
___________
9x + 22
+ 9x^2
- 45x
___________
67x + 22
The quotient is 2x - 3, and the remainder is 67x + 22. Hence, none of the given options (1, 2, 3, or 4) is correct.