Picture Frame Area Polynomial Expression And Increase Calculation

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Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving a picture frame and some polynomial expressions. Get ready to dust off your algebra skills as we explore how to calculate the area of this frame and how it changes when we increase its dimensions. Let's get started!

a) Crafting a Polynomial Expression for the Frame's Area

Our initial task revolves around determining the area of the picture frame, which has dimensions 4x - 6 by 3x + 5. To tackle this, we'll employ our knowledge of area calculation for rectangles, which states that the area is simply the product of its length and width. In our case, this translates to multiplying the two given dimensions: (4x - 6) and (3x + 5).

To effectively multiply these binomials, we'll utilize the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break it down step by step:

  • First: Multiply the first terms of each binomial: 4x * 3x = 12x²
  • Outer: Multiply the outer terms of the binomials: 4x * 5 = 20x
  • Inner: Multiply the inner terms of the binomials: -6 * 3x = -18x
  • Last: Multiply the last terms of each binomial: -6 * 5 = -30

Now, let's combine these results:

12x² + 20x - 18x - 30

Our final step involves simplifying the expression by combining like terms. In this case, we can combine the 20x and -18x terms:

12x² + (20x - 18x) - 30 = 12x² + 2x - 30

And there you have it! The polynomial expression representing the area of the picture frame is 12x² + 2x - 30. This expression elegantly captures how the area of the frame changes as the value of x varies. Remember, guys, understanding polynomial expressions is crucial for solving many real-world problems, and this picture frame example is a great illustration of that.

This expression tells us a lot about how the area of the frame changes based on the value of x. For instance, if x is a small number, the 12x² term will be relatively small, and the area will be more influenced by the 2x and -30 terms. However, as x gets larger, the 12x² term will dominate, and the area will increase much more rapidly. It's the beauty of polynomials, they allow us to model complex relationships in a concise and powerful way.

b) Gauging the Area Increase After Dimension Augmentation

Now, let's tackle the second part of our problem. We're tasked with figuring out how much the area increases when each dimension of the frame is increased by 2x. This adds another layer of algebraic fun to our challenge. So, stick with me, guys, we will make it through this.

First, we need to determine the new dimensions of the frame after the increase. If the original dimensions were 4x - 6 and 3x + 5, adding 2x to each gives us:

  • New length: (4x - 6) + 2x = 6x - 6
  • New width: (3x + 5) + 2x = 5x + 5

With the new dimensions in hand, we can calculate the new area using the same principle as before: multiplying the length and the width. So, we'll multiply (6x - 6) by (5x + 5):

Again, we'll employ the distributive property (FOIL) to expand this product:

  • First: 6x * 5x = 30x²
  • Outer: 6x * 5 = 30x
  • Inner: -6 * 5x = -30x
  • Last: -6 * 5 = -30

Combining these terms gives us:

30x² + 30x - 30x - 30

Simplifying by combining like terms (in this case, the 30x and -30x terms cancel each other out), we get the expression for the new area:

30x² - 30

Now comes the crucial step: determining the increase in area. To do this, we'll subtract the original area (which we found in part a to be 12x² + 2x - 30) from the new area (30x² - 30):

(30x² - 30) - (12x² + 2x - 30)

Remember, when subtracting a polynomial expression, we need to distribute the negative sign to each term inside the parentheses:

30x² - 30 - 12x² - 2x + 30

Finally, we combine like terms to simplify the expression:

(30x² - 12x²) - 2x + (-30 + 30) = 18x² - 2x

Therefore, the amount the area increased by is 18x² - 2x. This polynomial expression tells us exactly how much larger the new frame is compared to the original.

This result is quite interesting. The increase in area, represented by 18x² - 2x, depends heavily on the value of x. The 18x² term indicates that the increase in area grows quadratically with x, meaning that as x gets larger, the increase in area becomes significantly larger. The -2x term, on the other hand, provides a linear decrease, but its impact is less pronounced as x increases. This highlights the dynamic relationship between the dimensions of the frame and its area.

In conclusion, we've successfully navigated through this mathematical exploration of a picture frame's area. We crafted a polynomial expression to represent the original area and then calculated the increase in area after the dimensions were augmented. This journey has not only reinforced our understanding of polynomial expressions but also showcased their practical applications in real-world scenarios. Keep practicing, guys, and you'll become math masters in no time! Remember that each step we take in algebra is a step towards building a stronger foundation for more advanced mathematical concepts. So embrace the challenge, and let's continue to unlock the wonders of mathematics together!