Multiply Polynomials Step-by-Step Guide To Simplifying (3x - 5)(-x + 4)

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In the realm of algebra, mastering the multiplication of polynomials is a fundamental skill. Polynomials, algebraic expressions consisting of variables and coefficients, form the building blocks of more complex equations and functions. This comprehensive guide delves into the step-by-step process of multiplying the binomials (3x - 5) and (-x + 4), ultimately simplifying the product into standard form. Understanding this process is crucial for students, educators, and anyone involved in mathematical disciplines. This article will provide a clear, detailed explanation, ensuring that you grasp the underlying principles and can apply them confidently.

Understanding the Distributive Property

At the heart of polynomial multiplication lies the distributive property, a cornerstone of algebraic manipulation. This property dictates how to multiply a sum (or difference) by a term. In essence, the term outside the parentheses is multiplied by each term inside the parentheses. To effectively tackle the multiplication of extbf{(3x - 5) and (-x + 4)}, we must meticulously apply the distributive property, ensuring every term in the first binomial interacts with every term in the second binomial. This methodical approach is key to avoiding errors and achieving accurate results. By understanding and applying the distributive property correctly, we lay the foundation for simplifying more complex polynomial expressions and solving a wide range of algebraic problems. Mastering this principle not only aids in polynomial multiplication but also in various other mathematical contexts, highlighting its importance in algebraic proficiency.

Step-by-Step Expansion of (3x - 5)(-x + 4)

To begin, we systematically apply the distributive property. First, we multiply each term in the first binomial (3x - 5) by each term in the second binomial (-x + 4). This involves four distinct multiplications:

  1. (3x) multiplied by (-x)
  2. (3x) multiplied by (4)
  3. (-5) multiplied by (-x)
  4. (-5) multiplied by (4)

This process ensures that every term in the first binomial interacts with every term in the second binomial, laying the groundwork for accurate simplification. Each multiplication is a crucial step, and performing them with care and precision is essential for achieving the correct final result. This initial expansion sets the stage for combining like terms and expressing the product in its simplest, standard form. The methodical application of the distributive property is the key to unraveling the product of these two binomials, transforming a complex expression into a manageable form for further simplification and analysis.

Performing the Multiplication

Following the distributive property, we perform each multiplication individually:

  1. (3x)(-x) = -3x^2: Here, the coefficients 3 and -1 are multiplied to yield -3, and the variable x is multiplied by itself, resulting in x². The product is therefore -3x².
  2. (3x)(4) = 12x: In this case, the coefficient 3 is multiplied by 4, giving 12, and the variable x remains unchanged. The result is 12x.
  3. (-5)(-x) = 5x: The negative sign of -5 and -x cancel each other out, resulting in a positive product. The coefficient 5 is multiplied by the implicit coefficient of 1 in front of x, yielding 5, and the variable x remains. Thus, the product is 5x.
  4. (-5)(4) = -20: Here, -5 is multiplied by 4, resulting in -20. There are no variables involved in this multiplication, so the product is a constant term.

By meticulously performing each of these multiplications, we break down the initial complex expression into a series of simpler terms. This step-by-step approach ensures accuracy and sets the stage for combining like terms, a crucial process in simplifying polynomial expressions. The careful execution of these individual multiplications is the cornerstone of correctly expanding and simplifying the product of the given binomials.

Combining the Terms

After applying the distributive property and performing the individual multiplications, we gather the resulting terms:

-3x^2 + 12x + 5x - 20

This expression now contains four terms, some of which can be combined to further simplify the polynomial. The next step involves identifying and combining like terms, which are terms that have the same variable raised to the same power. In this expression, we have two like terms involving the variable x, namely 12x and 5x. Combining like terms is a fundamental algebraic technique that simplifies expressions and makes them easier to work with. By accurately combining like terms, we move closer to expressing the polynomial in its standard form, which is a crucial step in solving equations and performing further algebraic manipulations. The ability to identify and combine like terms efficiently is a key skill in algebraic simplification and problem-solving.

Simplifying and Expressing in Standard Form

In our expression, -3x^2 + 12x + 5x - 20, we identify 12x and 5x as like terms. Combining these terms, we add their coefficients: 12 + 5 = 17. Thus, 12x + 5x simplifies to 17x. Now, we substitute this simplified term back into our expression, resulting in:

-3x^2 + 17x - 20

This expression is now in standard form, which means the terms are arranged in descending order of their exponents. The standard form of a polynomial is written with the term containing the highest power of the variable first, followed by terms with lower powers, and finally the constant term. In this case, the term with x² comes first, followed by the term with x, and then the constant term. Expressing polynomials in standard form is essential for several reasons. It provides a consistent and organized way to represent polynomials, making them easier to compare and manipulate. Standard form is also crucial for identifying the degree and leading coefficient of the polynomial, which are important characteristics used in various algebraic operations and analyses. Therefore, ensuring a polynomial is in standard form is a fundamental step in algebraic problem-solving.

Final Answer: The Simplified Product

Therefore, the simplified product of (3x - 5)(-x + 4) in standard form is:

-3x^2 + 17x - 20

This final result is a quadratic expression, characterized by the highest power of the variable x being 2. The coefficients of the terms are -3, 17, and -20, representing the quadratic, linear, and constant terms, respectively. This simplified form is not only easier to understand and work with but also provides valuable insights into the behavior and properties of the original expression. The process of expanding and simplifying polynomial expressions is a cornerstone of algebra, enabling us to solve equations, graph functions, and tackle more complex mathematical problems. The ability to confidently multiply and simplify polynomials is a fundamental skill that opens doors to advanced algebraic concepts and applications.

Conclusion: Mastering Polynomial Multiplication

In conclusion, multiplying polynomials like (3x - 5)(-x + 4) involves a systematic application of the distributive property, careful multiplication of individual terms, combining like terms, and arranging the final expression in standard form. This process is a cornerstone of algebraic manipulation and is essential for solving equations, simplifying expressions, and understanding more advanced mathematical concepts. By mastering these techniques, students and practitioners alike can confidently navigate the world of algebra and beyond. The ability to accurately and efficiently multiply polynomials is not just a mathematical skill; it is a tool for problem-solving and critical thinking that extends to various fields of study and real-world applications. Whether you are a student learning the basics or a professional applying mathematical principles, a solid understanding of polynomial multiplication is invaluable.