Q) Given That The Volume Of A Cuboid Is Given By The Expression Volume = Length × Breadth X Height. Now If The Volume Of A Cuboid Is Expressed By The Polynomial X³ + 13x² + 32x + 20, Find The Expression For Length, Breadth And Height In Terms Of X, If
Introduction
In mathematics, the volume of a cuboid is given by the expression Volume = length × breadth × height. However, when the volume is expressed as a polynomial, it can be challenging to find the individual expressions for length, breadth, and height. In this article, we will explore how to find the expression for length, breadth, and height in terms of x, given that the volume of a cuboid is expressed by the polynomial x³ + 13x² + 32x + 20.
Understanding the Problem
The problem requires us to find the expression for length, breadth, and height in terms of x, given that the volume of a cuboid is expressed by the polynomial x³ + 13x² + 32x + 20. To do this, we need to factorize the polynomial and express it as a product of three linear factors.
Factorizing the Polynomial
The given polynomial is x³ + 13x² + 32x + 20. To factorize this polynomial, we can use the method of grouping. We can group the first two terms and the last two terms as follows:
x³ + 13x² + 32x + 20 = (x³ + 13x²) + (32x + 20)
Now, we can factor out the common term x² from the first group and the common term 4 from the second group:
x²(x + 13) + 4(8x + 5)
Next, we can factor out the common term (x + 13) from the first group and the common term (8x + 5) from the second group:
x²(x + 13) + 4(8x + 5) = (x + 13)(x² + 4(8x + 5))
Now, we can simplify the expression further by expanding the second group:
(x + 13)(x² + 4(8x + 5)) = (x + 13)(x² + 32x + 20)
Expressing the Volume as a Product of Three Linear Factors
Now that we have factorized the polynomial, we can express the volume as a product of three linear factors. The factorized form of the polynomial is (x + 13)(x² + 32x + 20). We can express this as a product of three linear factors as follows:
(x + 13)(x² + 32x + 20) = (x + 13)(x + 4)(x + 5)
Finding the Expression for Length, Breadth, and Height
Now that we have expressed the volume as a product of three linear factors, we can find the expression for length, breadth, and height. The three linear factors are (x + 13), (x + 4), and (x + 5). We can assign these factors to the length, breadth, and height as follows:
Length = x + 13 Breadth = x + 4 Height = x + 5
Conclusion
In this article, we have explored how to find the expression for length, breadth, and height in terms of x, given that the volume of a cuboid is expressed by the polynomial x³ + 13x² + 32x + 20. We have factorized the polynomial and expressed it as a product of three linear factors. We have then assigned these factors to the length, breadth, and height to find the expression for each. The expressions for length, breadth, and height are x + 13, x + 4, and x + 5, respectively.
Example Use Case
The expression for length, breadth, and height can be used to find the volume of a cuboid with a given length, breadth, and height. For example, if the length, breadth, and height of a cuboid are 10, 5, and 3, respectively, we can find the volume by substituting these values into the expression for volume:
Volume = length × breadth × height = (10 + 13)(5 + 4)(3 + 5) = 23 × 9 × 8 = 1656
Step-by-Step Solution
To find the expression for length, breadth, and height, follow these steps:
- Factorize the polynomial using the method of grouping.
- Express the factorized form as a product of three linear factors.
- Assign the three linear factors to the length, breadth, and height.
- Use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height.
Common Mistakes
When finding the expression for length, breadth, and height, common mistakes include:
- Not factorizing the polynomial correctly.
- Not expressing the factorized form as a product of three linear factors.
- Not assigning the three linear factors to the length, breadth, and height correctly.
- Not using the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height.
Conclusion
Q: What is the main concept behind finding the expression for length, breadth, and height of a cuboid from a polynomial volume?
A: The main concept behind finding the expression for length, breadth, and height of a cuboid from a polynomial volume is to factorize the polynomial and express it as a product of three linear factors. This allows us to assign the three linear factors to the length, breadth, and height, and use them to find the volume of a cuboid with a given length, breadth, and height.
Q: How do I factorize a polynomial to find the expression for length, breadth, and height?
A: To factorize a polynomial, you can use the method of grouping. This involves grouping the first two terms and the last two terms, and then factorizing out the common terms. You can also use other methods such as factoring by difference of squares or factoring by grouping.
Q: What are the common mistakes to avoid when finding the expression for length, breadth, and height?
A: Some common mistakes to avoid when finding the expression for length, breadth, and height include:
- Not factorizing the polynomial correctly.
- Not expressing the factorized form as a product of three linear factors.
- Not assigning the three linear factors to the length, breadth, and height correctly.
- Not using the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height.
Q: How do I use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height?
A: To use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height, you can substitute the given values into the expressions for length, breadth, and height, and then multiply the results together.
Q: What is the significance of finding the expression for length, breadth, and height of a cuboid from a polynomial volume?
A: Finding the expression for length, breadth, and height of a cuboid from a polynomial volume is significant because it allows us to find the volume of a cuboid with a given length, breadth, and height. This is useful in a variety of applications, such as architecture, engineering, and design.
Q: Can I use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are not integers?
A: Yes, you can use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are not integers. Simply substitute the given values into the expressions for length, breadth, and height, and then multiply the results together.
Q: How do I handle complex expressions for length, breadth, and height?
A: To handle complex expressions for length, breadth, and height, you can use algebraic techniques such as simplifying the expressions, combining like terms, and using the distributive property.
Q: Can I use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are negative?
A: Yes, you can use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are negative. Simply substitute the given values into the expressions for length, breadth, and height, and then multiply the results together.
Q: What are some real-world applications of finding the expression for length, breadth, and height of a cuboid from a polynomial volume?
A: Some real-world applications of finding the expression for length, breadth, and height of a cuboid from a polynomial volume include:
- Architecture: Finding the volume of a building with a given length, breadth, and height.
- Engineering: Finding the volume of a machine part with a given length, breadth, and height.
- Design: Finding the volume of a product with a given length, breadth, and height.
Q: Can I use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are fractions?
A: Yes, you can use the expressions for length, breadth, and height to find the volume of a cuboid with a given length, breadth, and height that are fractions. Simply substitute the given values into the expressions for length, breadth, and height, and then multiply the results together.
Q: How do I verify the accuracy of the expressions for length, breadth, and height?
A: To verify the accuracy of the expressions for length, breadth, and height, you can use algebraic techniques such as simplifying the expressions, combining like terms, and using the distributive property. You can also use numerical methods such as plugging in values to check the accuracy of the expressions.