IDNLearn.com makes it easy to find the right answers to your questions. Discover in-depth and reliable answers to all your questions from our knowledgeable community members who are always ready to assist.
Sagot :
Let's analyze the given quotient step-by-step.
Given:
[tex]\[ \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} \][/tex]
### Step 1: Simplify Each Fraction
First, simplify each fraction separately.
1. Simplify [tex]\(\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7}\)[/tex]:
- Numerator: [tex]\(3 x^2 - 27 x\)[/tex]
- Denominator: [tex]\(2 x^2 + 13 x - 7\)[/tex]
Factorize the numerator and denominator if possible:
- Numerator: [tex]\(3x(x - 9)\)[/tex]
- Denominator: This is a bit too complex for manual factorization here, so we will simplify it directly using algebraic techniques revealing simplified form stays as it is.
2. Simplify [tex]\(\frac{3 x}{4 x^2 - 1}\)[/tex]:
- Numerator: [tex]\(3 x\)[/tex]
- Denominator: [tex]\(4 x^2 - 1\)[/tex]
Factorize the denominator:
- Denominator: [tex]\((2x - 1)(2x + 1)\)[/tex]
### Step 2: Divide the Simplified Fractions
Dividing the two fractions is equivalent to multiplying the first fraction by the reciprocal of the second.
[tex]\[ \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} = \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \times \frac{4 x^2 - 1}{3 x} \][/tex]
Combine into a single fraction:
[tex]\[ \frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
### Step 3: Simplify the Resulting Expression
Simplify the fraction:
[tex]\[ \frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
Factorize [tex]\(3 x^2 - 27 x\)[/tex] as [tex]\(3x(x - 9)\)[/tex].
Rewrite the entire fraction:
[tex]\[ \frac{3x(x - 9) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
Cancel out terms common to numerator and denominator:
[tex]\[ \frac{(x - 9) \cdot (4 x^2 - 1)}{2 x^2 + 13 x - 7} \][/tex]
Notice the denominator [tex]\(4 x^2 - 1\)[/tex] can be written as [tex]\((2x - 1)(2x + 1)\)[/tex]:
Simplify further:
[tex]\[ = \frac{(x - 9) \cdot (2x - 1)(2x + 1)}{2 x^2 + 13 x - 7} \][/tex]
### Step 4: Simplify to the Most Reduced Form
Combine and further reduce any additional common factors which simplification gives:
### Result
As we solved the simplification, we found the final form of numerator and denominator are :
[tex]\(\boxed{2x^2 - 17x - 9}\)[/tex]
and
[tex]\(\boxed{x + 7}\)[/tex]
### Step 5: Identify Non-existent Points
Finally, determine when the expression does not exist, which occurs when the denominator is zero.
[tex]\[ x + 7 = 0 \implies x = -7 \][/tex]
Thus, the expression does not exist for [tex]\(x = \boxed{-7}\)[/tex].
So, the simplest form of the quotient is:
- Numerator: [tex]\(2x^2 - 17x - 9\)[/tex]
- Denominator: [tex]\(x + 7\)[/tex]
- The expression does not exist when [tex]\(x = -7\)[/tex].
Given:
[tex]\[ \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} \][/tex]
### Step 1: Simplify Each Fraction
First, simplify each fraction separately.
1. Simplify [tex]\(\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7}\)[/tex]:
- Numerator: [tex]\(3 x^2 - 27 x\)[/tex]
- Denominator: [tex]\(2 x^2 + 13 x - 7\)[/tex]
Factorize the numerator and denominator if possible:
- Numerator: [tex]\(3x(x - 9)\)[/tex]
- Denominator: This is a bit too complex for manual factorization here, so we will simplify it directly using algebraic techniques revealing simplified form stays as it is.
2. Simplify [tex]\(\frac{3 x}{4 x^2 - 1}\)[/tex]:
- Numerator: [tex]\(3 x\)[/tex]
- Denominator: [tex]\(4 x^2 - 1\)[/tex]
Factorize the denominator:
- Denominator: [tex]\((2x - 1)(2x + 1)\)[/tex]
### Step 2: Divide the Simplified Fractions
Dividing the two fractions is equivalent to multiplying the first fraction by the reciprocal of the second.
[tex]\[ \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} = \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \times \frac{4 x^2 - 1}{3 x} \][/tex]
Combine into a single fraction:
[tex]\[ \frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
### Step 3: Simplify the Resulting Expression
Simplify the fraction:
[tex]\[ \frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
Factorize [tex]\(3 x^2 - 27 x\)[/tex] as [tex]\(3x(x - 9)\)[/tex].
Rewrite the entire fraction:
[tex]\[ \frac{3x(x - 9) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x} \][/tex]
Cancel out terms common to numerator and denominator:
[tex]\[ \frac{(x - 9) \cdot (4 x^2 - 1)}{2 x^2 + 13 x - 7} \][/tex]
Notice the denominator [tex]\(4 x^2 - 1\)[/tex] can be written as [tex]\((2x - 1)(2x + 1)\)[/tex]:
Simplify further:
[tex]\[ = \frac{(x - 9) \cdot (2x - 1)(2x + 1)}{2 x^2 + 13 x - 7} \][/tex]
### Step 4: Simplify to the Most Reduced Form
Combine and further reduce any additional common factors which simplification gives:
### Result
As we solved the simplification, we found the final form of numerator and denominator are :
[tex]\(\boxed{2x^2 - 17x - 9}\)[/tex]
and
[tex]\(\boxed{x + 7}\)[/tex]
### Step 5: Identify Non-existent Points
Finally, determine when the expression does not exist, which occurs when the denominator is zero.
[tex]\[ x + 7 = 0 \implies x = -7 \][/tex]
Thus, the expression does not exist for [tex]\(x = \boxed{-7}\)[/tex].
So, the simplest form of the quotient is:
- Numerator: [tex]\(2x^2 - 17x - 9\)[/tex]
- Denominator: [tex]\(x + 7\)[/tex]
- The expression does not exist when [tex]\(x = -7\)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.