Simplifying (3x+4)(2x-5)-11x(x-2)+5(x^2-3x-1) To An Integer
In mathematics, simplifying complex expressions is a fundamental skill. This article delves into the simplification of the algebraic expression (3x+4)(2x-5)-11x(x-2)+5(x^2-3x-1), aiming to demonstrate that, after simplification, it results in an integer. This type of problem often appears in introductory algebra courses and serves as a good exercise in polynomial manipulation, distribution, and combining like terms. Let's embark on this mathematical journey, breaking down each step to reveal the final, constant value.
Expanding the Expression
To begin simplifying the expression, we first need to expand each term using the distributive property (also known as the FOIL method for binomials). This involves multiplying each term inside the parentheses by the terms outside. Starting with the first term, (3x+4)(2x-5), we multiply 3x by both 2x and -5, and then multiply 4 by both 2x and -5. This yields:
(3x * 2x) + (3x * -5) + (4 * 2x) + (4 * -5) = 6x^2 - 15x + 8x - 20
Next, we expand the second term, -11x(x-2). Here, we distribute -11x to both x and -2:
(-11x * x) + (-11x * -2) = -11x^2 + 22x
Finally, we expand the third term, 5(x^2-3x-1), by distributing 5 to each term inside the parentheses:
(5 * x^2) + (5 * -3x) + (5 * -1) = 5x^2 - 15x - 5
By expanding each term, we transform the original expression into a more manageable form, setting the stage for the next step: combining like terms.
Combining Like Terms
After expanding the expression, the next crucial step is to combine like terms. This involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. From the previous expansions, we have:
6x^2 - 15x + 8x - 20 - 11x^2 + 22x + 5x^2 - 15x - 5
Let's group the like terms together: terms with x^2, terms with x, and the constant terms:
(6x^2 - 11x^2 + 5x^2) + (-15x + 8x + 22x - 15x) + (-20 - 5)
Now, we add the coefficients of each group:
- For x^2 terms: 6 - 11 + 5 = 0, so the x^2 term vanishes.
- For x terms: -15 + 8 + 22 - 15 = 0, so the x term also vanishes.
- For constant terms: -20 - 5 = -25
Combining these results, we get:
0x^2 + 0x - 25
This simplifies to -25, which is a constant integer. The vanishing of the variable terms demonstrates a key property of this expression, leading to a fixed numerical result regardless of the value of x.
The Result: An Integer
Through the process of expanding and combining like terms, we have successfully shown that the original expression simplifies to a constant integer, specifically -25. This outcome highlights the elegance of algebraic manipulation, where seemingly complex expressions can be reduced to simple, concrete values. The fact that the variable x disappears during the simplification process indicates that the expression's value is independent of x; it remains constant no matter the input value for x. This result is not only a testament to the power of algebraic simplification but also provides a useful insight into the nature of the expression itself.
Importance in Mathematics
Simplifying algebraic expressions to integers or simpler forms is a fundamental concept in mathematics. It is crucial for several reasons:
- Problem-Solving: Many mathematical problems, especially in algebra and calculus, require simplifying expressions to find solutions. A simplified expression makes it easier to solve equations, find derivatives, or compute integrals.
- Understanding Relationships: Simplification can reveal underlying relationships between variables and constants. In this case, it showed that the value of the expression is independent of x.
- Efficiency: Simplified expressions are easier to work with. They reduce the chance of errors in subsequent calculations and make the overall mathematical process more efficient.
- Mathematical Proofs: In mathematical proofs, simplifying expressions is often a necessary step to reach a conclusion or demonstrate a theorem.
- Applications in Other Fields: Mathematical simplification is used extensively in fields such as physics, engineering, computer science, and economics to model and solve real-world problems.
Therefore, mastering the techniques of algebraic simplification, such as distribution and combining like terms, is essential for anyone studying mathematics or related fields. This skill not only helps in solving specific problems but also builds a deeper understanding of mathematical structures and relationships.
Conclusion
In summary, we have demonstrated that the expression (3x+4)(2x-5)-11x(x-2)+5(x^2-3x-1) simplifies to the integer -25. This was achieved through the systematic application of algebraic principles, including the distributive property and the combination of like terms. The result underscores the importance of simplification in mathematics, as it transforms a complex expression into a clear and concise value. Moreover, this exercise serves as a practical illustration of how algebraic manipulation can reveal inherent properties of mathematical expressions, showcasing the elegance and power of mathematical reasoning. The ability to simplify expressions is a core skill that benefits students and professionals alike, enabling more efficient and effective problem-solving across various disciplines.
