Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![**Understanding Slant Asymptotes**
**Problem:**
Choose the correct slant asymptote for the function.
\[ f(x) = \frac{x^2 - 5x + 6}{x - 2} \]
**Options:**
a) \( y = x + 2 \)
b) \( y = x - 2 \)
c) \( y = x + 3 \)
d) \( y = x - 3 \)
**Solution:**
A slant (oblique) asymptote occurs when the degree of the polynomial in the numerator is one more than the degree of the polynomial in the denominator.
To find the slant asymptote of the given function
\[ f(x) = \frac{x^2 - 5x + 6}{x - 2} \]
we perform polynomial long division:
1. Divide the first term of the numerator by the first term of the denominator: \(\frac{x^2}{x} = x\).
2. Multiply the entire denominator by this term: \( x(x - 2) = x^2 - 2x \).
3. Subtract this product from the original numerator:
\[
(x^2 - 5x + 6) - (x^2 - 2x) = -3x + 6
\]
4. Repeat the process with the new polynomial:
\(\frac{-3x}{x} = -3\).
5. Multiply the entire denominator by this term: \(-3(x - 2) = -3x + 6\).
6. Subtract this product:
\[
(-3x + 6) - (-3x + 6) = 0
\]
Hence, the long division yields
\[ f(x) = x - 3 + \frac{0}{x - 2} \]
where the quotient \( y = x - 3 \) is the slant asymptote.
**Answer:**
d) \( y = x - 3 \)
This problem illustrates the process of finding slant asymptotes using polynomial division and helps to understand their behavior for certain rational functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefebba2e-ab3a-49b5-a7d3-0c6a95c353e0%2F6bcd3c79-570b-4545-90c8-aabe2f0dc554%2F06qlfw_processed.jpeg&w=3840&q=75)
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