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Magazine Chapter 8: Problem 85
Factor expression. \(81 c^{4} d^{4}-16 t^{4}\)
Short Answer
Expert verified
The expression factors to \((9c^2d^2 + 4t^2)(3cd + 2t)(3cd - 2t)\).
Step by step solution
01
Identify the Type of Expression
The expression \(81 c^4 d^4 - 16 t^4\) is a difference of squares. A difference of squares is in the form \(a^2 - b^2\).
02
Express as a Difference of Squares
Write each term as a perfect square: \( (9c^2d^2)^2 - (4t^2)^2 \). Here, \(9c^2d^2\) is \(a\) and \(4t^2\) is \(b\).
03
Apply the Difference of Squares Formula
Use the formula \(a^2 - b^2 = (a + b)(a - b)\) to factor the expression. Substitute \(a = 9c^2d^2\) and \(b = 4t^2\): \((9c^2d^2 + 4t^2)(9c^2d^2 - 4t^2)\).
04
Check for Further Factorization
Look at each factor: - \(9c^2d^2 + 4t^2\) cannot be factored further as it is the sum of squares.- \(9c^2d^2 - 4t^2\) is again a difference of squares because \((3cd)^2 - (2t)^2\).
05
Further Factor the Difference of Squares
The term \(9c^2d^2 - 4t^2\) can be factored using the difference of squares formula with \( (3cd)^2 - (2t)^2 \): \((3cd + 2t)(3cd - 2t)\).
06
Write the Final Factored Expression
Combine the factors from all the steps: \((9c^2d^2 + 4t^2)(3cd + 2t)(3cd - 2t)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
The concept of "Difference of Squares" is a cornerstone in algebraic factoring, making it easier to solve and simplify expressions. It occurs when you have an expression in the form \(a^2 - b^2\). This format is particularly special because it can be factored into a product of two binomials: \((a + b)(a - b)\).
This technique is powerful because it transforms a seemingly complex polynomial into simpler components that are more manageable. Consider the expression \(81c^4d^4 - 16t^4\).
We can view it as the difference of two perfect squares: \((9c^2d^2)^2 - (4t^2)^2\), where \(9c^2d^2\) and \(4t^2\) are perfect squares representing \(a\) and \(b\) respectively.
This technique is powerful because it transforms a seemingly complex polynomial into simpler components that are more manageable. Consider the expression \(81c^4d^4 - 16t^4\).
We can view it as the difference of two perfect squares: \((9c^2d^2)^2 - (4t^2)^2\), where \(9c^2d^2\) and \(4t^2\) are perfect squares representing \(a\) and \(b\) respectively.
- This helps in reducing the complexity of problems.
- Allows easier solutions and further factorizations.
- Efficiently used in simplifying algebraic expressions in algebra and calculus.
Algebraic Expressions
Algebraic expressions are a form of language in mathematics that includes numbers, variables, and operational symbols. These expressions can be simple, like \(2x + 3\), or more complex as shown in our exercise: \(81c^4d^4 - 16t^4\).
The idea here is to represent real-world problems in a standardized mathematical form, making solutions universal and applicable broadly. In the given problem, we identify components of algebraic expressions:
The idea here is to represent real-world problems in a standardized mathematical form, making solutions universal and applicable broadly. In the given problem, we identify components of algebraic expressions:
- **Variables:** Represent unknown values and are typically denoted by letters such as \(c, d, t\).
- **Coefficients:** Numbers that multiply the variables, like 81 and 16 in our example.
- **Exponents/Powers:** Indicate how many times a number or variable is multiplied by itself, as shown in the powers of 4.
Mathematics Education
Mathematics education is foundational for developing problem-solving and critical thinking skills. Teaching concepts like algebra and factoring cultivates logical reasoning applicable in real-life situations.
In educational settings, topics such as the "Difference of Squares" and "Algebraic Expressions" serve multiple purposes:
In educational settings, topics such as the "Difference of Squares" and "Algebraic Expressions" serve multiple purposes:
- **Foundation Building:** Provides a basis for higher mathematics courses and various applied sciences.
- **Problem Solving Skills:** Encourages students to break down complex problems into manageable chunks, making them easier to solve.
- **Logical Reasoning:** Enhances the ability to think logically and deduce equations efficiently.
