Question

Find the quotient and simplify (40x⁴)/(x²-1)÷(x¹⁰)/((x+1)²)

Answer 1

The quotient is:

`\[ (40x^4)/(x^2-1) / (x^(10))/((x+1)^2) = (40x^2)/((x-1)(x+1)(x+1)) \]`

Step-by-step explanation:

The given expression involves dividing a fraction by another fraction. To simplify this expression, we can start by factoring each term. The numerator,
`\(40x^4\), and the denominator, \(x^(10)\),` can be simplified by dividing both by the common factor
`\(x^2\), resulting in \(40\)`. Now, we have
`\((40)/((x^2-1)) / (1)/((x+1)^2)\).`

Next, we factor the denominator
`\(x^2-1\) as \((x-1)(x+1)\)`. The expression now becomes
`\((40)/((x-1)(x+1)) / (1)/((x+1)^2)\).` To divide by a fraction, we multiply by its reciprocal. Therefore, the expression is equivalent to
`\((40)/((x-1)(x+1)) * ((x+1)^2)/(1)\).`

Now, we can cancel out the common factor
`\((x+1)\)` in the numerator and denominator, resulting in the simplified expression
`\((40x^2)/((x-1)(x+1))\).` This is the final answer, representing the qu
`\((40)/((x^2-1)) / (1)/((x+1)^2)\).`otient of the given expression after simplification. The factors of
`\((x-1)\) and \((x+1)\)` in the denominator prevent division by zero, ensuring that the expression is defined for all real values of
`\(x\) except \(x = \pm 1\).`