Product Of Binomials Explained Solving (-3s + 2t)(4s - T)

In the realm of algebra, understanding how to multiply binomials is a fundamental skill. Binomials, which are algebraic expressions consisting of two terms, often appear in various mathematical contexts, from solving equations to simplifying complex expressions. In this article, we will delve into the specific product of two binomials: (-3s + 2t)(4s - t). Our aim is not just to find the answer, but to thoroughly understand the process involved, the underlying principles, and the significance of such operations in mathematics. By breaking down each step and explaining the concepts in detail, we will ensure that readers gain a comprehensive grasp of this topic.

The initial step in simplifying the product (-3s + 2t)(4s - t) involves applying 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, systematically expanding the expression. This methodical approach is crucial for avoiding errors and ensuring accuracy in algebraic manipulations. The distributive property is a cornerstone of algebra, allowing us to handle complex expressions by breaking them down into simpler, manageable parts. Applying FOIL correctly not only provides the solution but also reinforces a methodical approach to problem-solving, which is a valuable skill in mathematics and beyond.

Now, let's apply the FOIL method to our problem. First, we multiply the First terms: (-3s) * (4s), which gives us -12s². This step involves multiplying the coefficients (-3 and 4) and the variables (s and s), following the rules of exponents. Understanding how to handle coefficients and variables correctly is essential for mastering algebraic expressions. The result, -12s², represents the first part of our expanded expression. Next, we multiply the Outer terms: (-3s) * (-t), resulting in +3st. Here, the product of two negative terms yields a positive term, a key concept in algebra. The term 'st' represents the product of the variables 's' and 't', highlighting the importance of keeping track of variables during multiplication. This step demonstrates how the interaction between negative signs and variable multiplication contributes to the final expression.

Continuing with the FOIL method, we now multiply the Inner terms: (2t) * (4s), which gives us +8st. This term, similar to the previous one, involves the product of variables 's' and 't', but with a different coefficient. The order of multiplication does not affect the result, as multiplication is commutative. Understanding this property allows us to rearrange terms if needed, making it easier to combine like terms later on. Finally, we multiply the Last terms: (2t) * (-t), resulting in -2t². This step involves multiplying the coefficients (2 and -1) and the variables (t and t), resulting in a squared term. The negative sign is crucial here, as it changes the overall sign of the term. Combining all these steps, we get the expanded expression: -12s² + 3st + 8st - 2t².

The expanded expression, -12s² + 3st + 8st - 2t², contains like terms that can be combined to simplify the expression further. Like terms are terms that have the same variables raised to the same powers. In our expression, the terms +3st and +8st are like terms because they both contain the variables 's' and 't' raised to the power of 1. Combining these like terms involves adding their coefficients while keeping the variable part the same. This process of combining like terms is essential for simplifying algebraic expressions and making them easier to work with. It's a fundamental step in algebra that allows us to reduce complex expressions to their simplest forms, making them more manageable and understandable.

In our expanded expression, -12s² + 3st + 8st - 2t², the like terms are +3st and +8st. To combine these terms, we add their coefficients: 3 + 8 = 11. Therefore, the combined term is +11st. This step simplifies our expression by reducing the number of terms, making it cleaner and easier to interpret. After combining like terms, the expression becomes -12s² + 11st - 2t². This simplified form is the result of our initial multiplication and subsequent simplification. It represents the product of the two binomials in its most concise form, ready for further use in mathematical operations or analysis. Understanding how to combine like terms is crucial for simplifying complex algebraic expressions and arriving at the correct final answer.

The final simplified expression, -12s² + 11st - 2t², represents the product of the two binomials (-3s + 2t) and (4s - t). This expression is a quadratic expression in two variables, 's' and 't'. It contains terms with different degrees, including squared terms (s² and t²) and a mixed term (st). Understanding the nature of this expression is crucial for further mathematical operations, such as solving equations or graphing functions. The coefficients of each term ( -12, 11, and -2) play a significant role in the behavior of the expression and its graphical representation. This final result showcases the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. It also demonstrates the importance of paying attention to detail and following the correct steps to ensure accuracy in mathematical calculations.

Understanding the product of binomials, as we've demonstrated with (-3s + 2t)(4s - t), is a foundational skill in algebra. The FOIL method, combined with the ability to combine like terms, provides a systematic approach to simplifying such expressions. The final result, -12s² + 11st - 2t², is a quadratic expression that can be further analyzed and used in various mathematical contexts. Mastering these concepts not only enhances one's algebraic proficiency but also provides a solid foundation for more advanced mathematical topics. The ability to manipulate algebraic expressions accurately and efficiently is crucial for success in mathematics and related fields. This detailed exploration of the product of two binomials highlights the importance of understanding each step and the underlying principles, ensuring a comprehensive grasp of algebraic concepts.

What is the result of the product of the expression (-3s + 2t) and (4s - t)?

Product of Binomials Explained Solving (-3s + 2t)(4s - t)