Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Sure, let's solve the problem step-by-step to find the simplest form of the given product and determine any excluded values.
We begin with the given product:
[tex]\[ \frac{x^2 - 3x - 10}{x^2 - 6x + 5} \cdot \frac{x - 2}{x - 5} \][/tex]
First, we factorize the expressions in both the numerator and the denominator.
### Step 1: Factorize the numerator and the denominator
Factorize [tex]\(x^2 - 3x - 10\)[/tex]:
[tex]\[ x^2 - 3x - 10 = (x - 5)(x + 2) \][/tex]
Factorize [tex]\(x^2 - 6x + 5\)[/tex]:
[tex]\[ x^2 - 6x + 5 = (x - 1)(x - 5) \][/tex]
Now, substitute the factored forms into the product:
[tex]\[ \frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5} \][/tex]
### Step 2: Simplify the expression
Combine the numerator and the denominator:
[tex]\[ \frac{(x - 5)(x + 2)(x - 2)}{(x - 1)(x - 5)(x - 5)} \][/tex]
Notice that the factor [tex]\((x - 5)\)[/tex] appears in both the numerator and the denominator. We can cancel one [tex]\((x - 5)\)[/tex]:
[tex]\[ \frac{(x + 2)(x - 2)}{(x - 1)(x - 5)} \][/tex]
The numerator [tex]\((x + 2)(x - 2)\)[/tex] simplifies to:
[tex]\[ x^2 - 4 \][/tex]
Thus, the simplified form is:
[tex]\[ \frac{x^2 - 4}{(x - 1)(x - 5)} \][/tex]
### Step 3: Identify excluded values
The expression is undefined where the denominator is zero. We have:
[tex]\[ (x - 1)(x - 5) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
### Final answer
Therefore, the simplest form of the given product is:
[tex]\[ \frac{x^2 - 4}{(x - 1)(x - 5)} \][/tex]
The expression has excluded values at:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
So, the correct answers for the given drop-down menus are:
The simplest form of this product has a numerator of [tex]\(x^2 - 4\)[/tex] and a denominator of [tex]\((x - 1)(x - 5)\)[/tex]. The expression has an excluded value of [tex]\(x = 5 \vee x = 1\)[/tex].
We begin with the given product:
[tex]\[ \frac{x^2 - 3x - 10}{x^2 - 6x + 5} \cdot \frac{x - 2}{x - 5} \][/tex]
First, we factorize the expressions in both the numerator and the denominator.
### Step 1: Factorize the numerator and the denominator
Factorize [tex]\(x^2 - 3x - 10\)[/tex]:
[tex]\[ x^2 - 3x - 10 = (x - 5)(x + 2) \][/tex]
Factorize [tex]\(x^2 - 6x + 5\)[/tex]:
[tex]\[ x^2 - 6x + 5 = (x - 1)(x - 5) \][/tex]
Now, substitute the factored forms into the product:
[tex]\[ \frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5} \][/tex]
### Step 2: Simplify the expression
Combine the numerator and the denominator:
[tex]\[ \frac{(x - 5)(x + 2)(x - 2)}{(x - 1)(x - 5)(x - 5)} \][/tex]
Notice that the factor [tex]\((x - 5)\)[/tex] appears in both the numerator and the denominator. We can cancel one [tex]\((x - 5)\)[/tex]:
[tex]\[ \frac{(x + 2)(x - 2)}{(x - 1)(x - 5)} \][/tex]
The numerator [tex]\((x + 2)(x - 2)\)[/tex] simplifies to:
[tex]\[ x^2 - 4 \][/tex]
Thus, the simplified form is:
[tex]\[ \frac{x^2 - 4}{(x - 1)(x - 5)} \][/tex]
### Step 3: Identify excluded values
The expression is undefined where the denominator is zero. We have:
[tex]\[ (x - 1)(x - 5) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
### Final answer
Therefore, the simplest form of the given product is:
[tex]\[ \frac{x^2 - 4}{(x - 1)(x - 5)} \][/tex]
The expression has excluded values at:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
So, the correct answers for the given drop-down menus are:
The simplest form of this product has a numerator of [tex]\(x^2 - 4\)[/tex] and a denominator of [tex]\((x - 1)(x - 5)\)[/tex]. The expression has an excluded value of [tex]\(x = 5 \vee x = 1\)[/tex].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
