Comparing Pen Prices And Algebraic Expressions Factorization And LCM

To effectively compare the prices of pen A and pen B, we need specific information regarding their individual prices. Without knowing the cost of each pen, a direct comparison is impossible. Price comparison is a fundamental aspect of purchasing decisions, whether in a store or online. It helps consumers make informed choices and find the best deals. In a real-world scenario, the price comparison could involve looking at different brands, models, or retailers. For instance, pen A might be a high-end fountain pen, while pen B could be a standard ballpoint pen. The features, quality, and brand reputation significantly influence the pricing. When analyzing prices, factors such as the pen's ink type, barrel material, and overall design come into play. A detailed price comparison should also consider factors like shipping costs, warranty, and return policies, especially when purchasing online. Furthermore, customer reviews and ratings can provide insights into the pen's value and performance relative to its price. In the absence of specific pricing information, the comparison remains hypothetical. Suppose pen A is priced at $10, and pen B is priced at $5. In that case, pen A is twice as expensive as pen B. Alternatively, if pen A costs $3 and pen B costs $7, then pen B is more expensive than pen A. This highlights the importance of having actual price figures to make a meaningful comparison. Understanding the market value and the specific features of each pen is crucial for making an educated purchase decision. Price comparison websites and tools can be useful resources, providing up-to-date price information from various retailers. These tools often include features to track price changes and set alerts for price drops, ensuring consumers get the best possible deals.

This section delves into two algebraic expressions, $x^2 - 4x + 23$ and $x^2 - 5x + 6$, focusing on factorization and finding the Least Common Multiple (LCM). These concepts are essential in algebra and have wide applications in mathematics and related fields. Understanding how to factorize expressions and determine their LCM is crucial for simplifying equations, solving problems, and working with more complex algebraic structures. Let's break down each expression and address the questions posed.

a. Factorizing the First Expression: $x^2 - 4x + 23$

Factorizing the expression $x^2 - 4x + 23$ involves finding two binomials that, when multiplied, yield the original quadratic expression. However, this particular quadratic expression presents a challenge because it does not factor neatly using integer coefficients. To determine if it can be factored, we can examine its discriminant. The discriminant, denoted as Δ, is calculated using the formula Δ = $b^2 - 4ac$, where a, b, and c are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. In this case, a = 1, b = -4, and c = 23. Plugging these values into the discriminant formula, we get:

Δ = $(-4)^2 - 4(1)(23)$ = $16 - 92$ = -76

Since the discriminant is negative (-76), the quadratic equation $x^2 - 4x + 23 = 0$ has no real roots. This means that the expression cannot be factored into two binomials with real coefficients. Therefore, the expression $x^2 - 4x + 23$ is considered prime or irreducible over the real numbers. Attempting to factor it using techniques such as splitting the middle term or completing the square will not result in factors with integer or simple fractional coefficients. The inability to factor this expression using standard methods highlights the importance of understanding the discriminant and its role in determining the nature of the roots of a quadratic equation. In summary, $x^2 - 4x + 23$ cannot be factored using real numbers, making it a prime quadratic expression. However, if we were to consider complex numbers, we could find complex roots and express the expression in factored form using complex coefficients.

b. Finding the L.C.M of the Two Expressions: $x^2 - 4x + 23$ and $x^2 - 5x + 6$

To find the Least Common Multiple (LCM) of two algebraic expressions, we first need to factorize each expression. We already established in part (a) that the expression $x^2 - 4x + 23$ cannot be factored using real numbers. Now, let’s factorize the second expression, $x^2 - 5x + 6$. To factorize $x^2 - 5x + 6$, we look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, the factorization of $x^2 - 5x + 6$ is:

$x^2 - 5x + 6$ = $(x - 2)(x - 3)$

Now that we have the factorized forms of both expressions, we can determine the LCM. The LCM is the smallest expression that is divisible by both given expressions. Since $x^2 - 4x + 23$ is irreducible, and $x^2 - 5x + 6$ factors into $(x - 2)(x - 3)$, the LCM of the two expressions is simply their product:

LCM($x^2 - 4x + 23$, $x^2 - 5x + 6$) = $(x^2 - 4x + 23)(x - 2)(x - 3)$

This is because there are no common factors between the two expressions. The first expression, being irreducible, does not share any factors with the second expression, which factors into two distinct linear terms. Thus, the LCM is obtained by multiplying the unique factors from both expressions. In general, when finding the LCM of algebraic expressions, it’s crucial to factorize each expression completely. If there are any common factors, the LCM will include each factor raised to the highest power that appears in any of the expressions. In this specific case, because one expression is irreducible and the other factors into linear terms, the LCM is simply the product of the two expressions. Understanding the concept of LCM is vital in various algebraic operations, such as adding or subtracting rational expressions, and in solving equations involving fractions. The ability to accurately determine the LCM ensures that these operations are performed correctly and efficiently.

In summary, the LCM of $x^2 - 4x + 23$ and $x^2 - 5x + 6$ is $(x^2 - 4x + 23)(x - 2)(x - 3)$.

In conclusion, the comparison of pen prices requires specific pricing information for each pen. Without these details, the comparison remains theoretical. In the algebraic section, we found that the expression $x^2 - 4x + 23$ is irreducible over real numbers, while $x^2 - 5x + 6$ factors into $(x - 2)(x - 3)$. The LCM of these two expressions is their product, $(x^2 - 4x + 23)(x - 2)(x - 3)$. Understanding these algebraic concepts is crucial for solving more complex mathematical problems.