short answer.Find the quotient.6x^3-5x^2-13x+13/2x+3
To find the quotient of the polynomial division, you can use the long division method. Here's how you can solve it step by step:
Step 1: Rearrange the terms to write the dividend and divisor in decreasing order of powers:
Dividend: 6x^3 - 5x^2 - 13x + 13
Divisor: 2x + 3
Step 2: Divide the first term of the dividend (6x^3) by the first term of the divisor (2x). The result will be the first term of the quotient:
Quotient: 3x^2
Step 3: Multiply the divisor (2x + 3) by the first term of the quotient (3x^2). Place the result below the dividend:
3x^2
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2x + 3 ) | 6x^3 - 5x^2 - 13x + 13
Step 4: Subtract the product from the dividend:
3x^2
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2x + 3 ) | 6x^3 - 5x^2 - 13x + 13
- (6x^3 + 9x^2)
Simplifying: - 5x^2 - 13x + 13 - (6x^3 + 9x^2)
= -5x^2 - 13x + 13 - 6x^3 - 9x^2
= -6x^3 - 14x^2 - 13x + 13
Step 5: Bring down the next term from the dividend, which is -13x:
3x^2 - 6
Step 6: Repeat steps 2-4 until all terms in the dividend are used:
3x^2 - 6
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2x + 3 ) | 6x^3 - 5x^2 - 13x + 13
- (6x^3 + 9x^2)
Simplifying: - 5x^2 - 13x + 13 - (6x^3 + 9x^2)
= -5x^2 - 13x + 13 - 6x^3 - 9x^2
= -6x^3 - 14x^2 - 13x + 13
Multiply: - 6x^3 - 14x^2
- ( - 6x^3 - 9x^2)
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Step 7: Repeat steps 5 and 6 with the new polynomial:
3x^2 - 6 - 14
Step 8: Repeat steps 2-4 until all terms in the dividend are used:
3x^2 - 6 - 14
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2x + 3 ) | 6x^3 - 5x^2 - 13x + 13
- (6x^3 + 9x^2)
Simplifying: - 5x^2 - 13x + 13 - (6x^3 + 9x^2)
= -5x^2 - 13x + 13 - 6x^3 - 9x^2
= -6x^3 - 14x^2 - 13x + 13
Multiply: - 6x^3 - 14x^2 - 13x
- ( - 6x^3 - 9x^2 )
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Step 9: Repeat steps 5 and 6 with the new polynomial:
3x^2 - 6 - 14 - 13x
Step 10: Repeat steps 2-4 until all terms in the dividend are used:
3x^2 - 6 - 14 - 13x
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2x + 3 ) | 6x^3 - 5x^2 - 13x + 13
- (6x^3 + 9x^2)
Simplifying: - 5x^2 - 13x + 13 - (6x^3 + 9x^2)
= -5x^2 - 13x + 13 - 6x^3 - 9x^2
= -6x^3 - 14x^2 - 13x + 13
Multiply: - 6x^3 - 14x^2 - 13x + 13
- ( - 6x^3 - 9x^2 )
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Step 11: There are no more terms in the dividend, so the remainder is 0.
Quotient: 3x^2 - 6 - 14 - 13x
Therefore, the quotient of (6x^3 - 5x^2 - 13x + 13) divided by (2x + 3) is:
3x^2 - 6 - 14 - 13x, or simply written as 3x^2 - 13x - 20.