Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine whether the expression [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] simplifies to [tex]\(x\)[/tex], let's conduct a step-by-step simplification of the expression.
### Step 1: Factor the Numerator and Denominator
First, we need to factor both the numerator ([tex]\(x^3 - 1\)[/tex]) and the denominator ([tex]\(x^2 - 1\)[/tex]).
Factor the numerator:
The expression [tex]\(x^3 - 1\)[/tex] can be factored using the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In our case, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \][/tex]
Factor the denominator:
The expression [tex]\(x^2 - 1\)[/tex] can be factored using the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In our case, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
### Step 2: Rewrite the Expression with Factored Forms
Substitute the factored forms of the numerator and denominator into the original expression:
[tex]\[ \frac{x^3 - 1}{x^2 - 1} = \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} \][/tex]
### Step 3: Simplify the Expression
We can now cancel the common factor [tex]\((x - 1)\)[/tex] in the numerator and the denominator (assuming [tex]\(x \neq 1\)[/tex]):
[tex]\[ \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} = \frac{x^2 + x + 1}{x + 1} \][/tex]
### Step 4: Examine the Simplified Expression
The simplified expression is [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex]. It does not simplify further to [tex]\(x\)[/tex].
To determine whether [tex]\(\frac{x^2 + x + 1}{x + 1} = x\)[/tex], you can examine it more closely:
For the expression [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex] to equal [tex]\(x\)[/tex]:
[tex]\[ x \neq -1 \][/tex]
both sides of the equation [tex]\(\frac{x^2 + x + 1}{x + 1} = x\)[/tex] should hold:
Set up the equation:
[tex]\[ \frac{x^2 + x + 1}{x + 1} = x \][/tex]
Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[ x^2 + x + 1 = x \cdot (x + 1) \][/tex]
[tex]\[ x^2 + x + 1 = x^2 + x \][/tex]
Subtract [tex]\(x^2 + x\)[/tex] from both sides:
[tex]\[ 1 = 0 \][/tex]
### Conclusion
Clearly, the above equation [tex]\(1 = 0\)[/tex] is a contradiction, which means [tex]\(\frac{x^2 + x + 1}{x + 1} \neq x\)[/tex].
So, [tex]\(\frac{x^3 - 1}{x^2 - 1} \neq x\)[/tex].
Therefore, the correct answer is:
No, because the simplified form of the expression [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] is [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex] and it doesn't simplify to [tex]\(x\)[/tex].
### Step 1: Factor the Numerator and Denominator
First, we need to factor both the numerator ([tex]\(x^3 - 1\)[/tex]) and the denominator ([tex]\(x^2 - 1\)[/tex]).
Factor the numerator:
The expression [tex]\(x^3 - 1\)[/tex] can be factored using the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In our case, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \][/tex]
Factor the denominator:
The expression [tex]\(x^2 - 1\)[/tex] can be factored using the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In our case, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
### Step 2: Rewrite the Expression with Factored Forms
Substitute the factored forms of the numerator and denominator into the original expression:
[tex]\[ \frac{x^3 - 1}{x^2 - 1} = \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} \][/tex]
### Step 3: Simplify the Expression
We can now cancel the common factor [tex]\((x - 1)\)[/tex] in the numerator and the denominator (assuming [tex]\(x \neq 1\)[/tex]):
[tex]\[ \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} = \frac{x^2 + x + 1}{x + 1} \][/tex]
### Step 4: Examine the Simplified Expression
The simplified expression is [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex]. It does not simplify further to [tex]\(x\)[/tex].
To determine whether [tex]\(\frac{x^2 + x + 1}{x + 1} = x\)[/tex], you can examine it more closely:
For the expression [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex] to equal [tex]\(x\)[/tex]:
[tex]\[ x \neq -1 \][/tex]
both sides of the equation [tex]\(\frac{x^2 + x + 1}{x + 1} = x\)[/tex] should hold:
Set up the equation:
[tex]\[ \frac{x^2 + x + 1}{x + 1} = x \][/tex]
Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[ x^2 + x + 1 = x \cdot (x + 1) \][/tex]
[tex]\[ x^2 + x + 1 = x^2 + x \][/tex]
Subtract [tex]\(x^2 + x\)[/tex] from both sides:
[tex]\[ 1 = 0 \][/tex]
### Conclusion
Clearly, the above equation [tex]\(1 = 0\)[/tex] is a contradiction, which means [tex]\(\frac{x^2 + x + 1}{x + 1} \neq x\)[/tex].
So, [tex]\(\frac{x^3 - 1}{x^2 - 1} \neq x\)[/tex].
Therefore, the correct answer is:
No, because the simplified form of the expression [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] is [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex] and it doesn't simplify to [tex]\(x\)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
