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Sagot :
To find the quotient of the given expression:
[tex]\[ \frac{\frac{2 y^2 - 6 y - 20}{4 y + 12}}{\frac{y^2 + 5 y + 6}{3 y^2 + 18 y + 27}} \][/tex]
we need to follow a step-by-step approach for simplification.
1. Factorization of the Numerator and Denominator:
First, let's factorize each part of the fractions involved.
- Numerator: [tex]\(2 y^2 - 6 y - 20\)[/tex]
[tex]\[ 2 y^2 - 6 y - 20 = 2(y^2 - 3y - 10) = 2(y - 5)(y + 2) \][/tex]
- Denominator: [tex]\(4 y + 12\)[/tex]
[tex]\[ 4 y + 12 = 4(y + 3) \][/tex]
- Numerator: [tex]\(y^2 + 5 y + 6\)[/tex]
[tex]\[ y^2 + 5 y + 6 = (y + 2)(y + 3) \][/tex]
- Denominator: [tex]\(3 y^2 + 18 y + 27\)[/tex]
[tex]\[ 3 y^2 + 18 y + 27 = 3(y^2 + 6y + 9) = 3(y + 3)^2 \][/tex]
Now, let's rewrite the original expression using these factorizations:
[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \div \frac{(y+2)(y+3)}{3(y+3)^2} \][/tex]
2. Division of Fractions:
To divide fractions, multiply by the reciprocal of the second fraction:
[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \times \frac{3(y+3)^2}{(y+2)(y+3)} \][/tex]
3. Simplification:
Before multiplying, cancel common factors:
- [tex]\((y + 2)\)[/tex] in the numerator and denominator.
- One [tex]\((y + 3)\)[/tex] factor in the numerator and denominator.
This reduces to:
[tex]\[ \frac{2(y-5)}{4} \times \frac{3(y+3)}{1} \][/tex]
- Simplify [tex]\(\frac{2(y - 5)}{4}\)[/tex]:
[tex]\[ \frac{2(y - 5)}{4} = \frac{1(y - 5)}{2} = \frac{y - 5}{2} \][/tex]
Now, multiply by [tex]\(3(y + 3)\)[/tex]:
[tex]\[ \frac{y - 5}{2} \times 3(y + 3) \][/tex]
Multiply the numerators and the denominators:
[tex]\[ \frac{3(y - 5)(y + 3)}{2} \][/tex]
This is the simplified form of the quotient.
Therefore, the correct answer is:
[tex]\[ \frac{3(y-5)(y+3)}{2} \][/tex]
However, we must match it with the given options. In this problem, it looks like none of the answer choices match directly our result:
None of the given answer options correctly matches the derived quotient [tex]\( \frac{3(y-5)(y+3)}{2} \)[/tex]. The answers may need checking for errors in the provided problem or options.
If re-evaluated step-by-step, the answer closest to our result seems mismatched. Confirm with the exact provided options if there might be a typographical or analytical adjustment possible.
[tex]\[ \frac{\frac{2 y^2 - 6 y - 20}{4 y + 12}}{\frac{y^2 + 5 y + 6}{3 y^2 + 18 y + 27}} \][/tex]
we need to follow a step-by-step approach for simplification.
1. Factorization of the Numerator and Denominator:
First, let's factorize each part of the fractions involved.
- Numerator: [tex]\(2 y^2 - 6 y - 20\)[/tex]
[tex]\[ 2 y^2 - 6 y - 20 = 2(y^2 - 3y - 10) = 2(y - 5)(y + 2) \][/tex]
- Denominator: [tex]\(4 y + 12\)[/tex]
[tex]\[ 4 y + 12 = 4(y + 3) \][/tex]
- Numerator: [tex]\(y^2 + 5 y + 6\)[/tex]
[tex]\[ y^2 + 5 y + 6 = (y + 2)(y + 3) \][/tex]
- Denominator: [tex]\(3 y^2 + 18 y + 27\)[/tex]
[tex]\[ 3 y^2 + 18 y + 27 = 3(y^2 + 6y + 9) = 3(y + 3)^2 \][/tex]
Now, let's rewrite the original expression using these factorizations:
[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \div \frac{(y+2)(y+3)}{3(y+3)^2} \][/tex]
2. Division of Fractions:
To divide fractions, multiply by the reciprocal of the second fraction:
[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \times \frac{3(y+3)^2}{(y+2)(y+3)} \][/tex]
3. Simplification:
Before multiplying, cancel common factors:
- [tex]\((y + 2)\)[/tex] in the numerator and denominator.
- One [tex]\((y + 3)\)[/tex] factor in the numerator and denominator.
This reduces to:
[tex]\[ \frac{2(y-5)}{4} \times \frac{3(y+3)}{1} \][/tex]
- Simplify [tex]\(\frac{2(y - 5)}{4}\)[/tex]:
[tex]\[ \frac{2(y - 5)}{4} = \frac{1(y - 5)}{2} = \frac{y - 5}{2} \][/tex]
Now, multiply by [tex]\(3(y + 3)\)[/tex]:
[tex]\[ \frac{y - 5}{2} \times 3(y + 3) \][/tex]
Multiply the numerators and the denominators:
[tex]\[ \frac{3(y - 5)(y + 3)}{2} \][/tex]
This is the simplified form of the quotient.
Therefore, the correct answer is:
[tex]\[ \frac{3(y-5)(y+3)}{2} \][/tex]
However, we must match it with the given options. In this problem, it looks like none of the answer choices match directly our result:
None of the given answer options correctly matches the derived quotient [tex]\( \frac{3(y-5)(y+3)}{2} \)[/tex]. The answers may need checking for errors in the provided problem or options.
If re-evaluated step-by-step, the answer closest to our result seems mismatched. Confirm with the exact provided options if there might be a typographical or analytical adjustment possible.
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