Expanding And Simplifying Polynomials A Step-by-Step Guide To (7x²) (2x³+5) (x²-4x-9)

Jul 11, 2025 by ADMIN 86 views

Unveiling the Product: Expanding and Simplifying (7x²) (2x³+5) (x²-4x-9)

In the realm of mathematics, algebraic expressions often present themselves as intricate puzzles waiting to be solved. One such puzzle is the expression (7x²) (2x³+5) (x²-4x-9), which appears complex at first glance. However, by systematically applying the principles of polynomial multiplication and simplification, we can unravel its structure and arrive at a concise and meaningful result. This article will delve into the step-by-step process of expanding and simplifying this expression, shedding light on the underlying mathematical concepts and techniques involved.

Breaking Down the Expression

To begin, let's dissect the expression (7x²) (2x³+5) (x²-4x-9) into its constituent parts. We have three factors: 7x², (2x³+5), and (x²-4x-9). The task at hand is to multiply these factors together to obtain a single polynomial expression. The order in which we multiply these factors doesn't affect the final result, thanks to the commutative property of multiplication. However, a strategic approach can often simplify the process.

One common strategy is to first multiply the two binomial factors, (2x³+5) and (x²-4x-9), and then multiply the result by the monomial factor, 7x². This approach allows us to manage the complexity of the expression in a stepwise manner. Let's embark on this journey of expansion and simplification.

Step 1: Multiplying the Binomial Factors

Our first task is to multiply the two binomial factors, (2x³+5) and (x²-4x-9). To do this, we'll employ the distributive property, which states that each term in the first factor must be multiplied by each term in the second factor. This process is often referred to as the FOIL method (First, Outer, Inner, Last) when multiplying two binomials. However, for factors with more than two terms, it's best to think of it as a systematic distribution.

Let's proceed with the multiplication:

(2x³+5) (x²-4x-9) = 2x³(x²-4x-9) + 5(x²-4x-9)

Now, we distribute 2x³ and 5 across the terms in the second factor:

= 2x³ * x² + 2x³ * (-4x) + 2x³ * (-9) + 5 * x² + 5 * (-4x) + 5 * (-9)

Next, we perform the individual multiplications:

= 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45

So, the product of the two binomial factors is 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45. This polynomial has six terms, each with a different power of x. Now, we move on to the next step: multiplying this result by the monomial factor, 7x².

Step 2: Multiplying by the Monomial Factor

Now that we have the product of the binomial factors, 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45, we need to multiply this polynomial by the monomial factor, 7x². Again, we'll use the distributive property, multiplying each term in the polynomial by 7x²:

7x² (2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45) = 7x² * 2x⁵ + 7x² * (-8x⁴) + 7x² * (-18x³) + 7x² * 5x² + 7x² * (-20x) + 7x² * (-45)

Performing the individual multiplications, we get:

= 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

This is the final expanded form of the original expression. It's a polynomial of degree 7, with terms ranging from x⁷ down to x². We have successfully navigated the multiplication and simplification process.

Step 3: Final Result and Simplification

After performing the multiplications, we arrive at the expanded form of the expression:

(7x²) (2x³+5) (x²-4x-9) = 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

This polynomial is already in its simplest form, as there are no like terms to combine. Like terms are terms that have the same variable raised to the same power. In this case, we have terms with x⁷, x⁶, x⁵, x⁴, x³, and x², each with a different exponent, so they cannot be combined.

Therefore, the final simplified expression is:

14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

This is the product of the original factors, expressed as a single polynomial. We have successfully expanded and simplified the given expression, unveiling its underlying structure.

Key Concepts and Techniques

Throughout this process, we've employed several key mathematical concepts and techniques. Let's recap them:

Distributive Property: This fundamental property allows us to multiply a factor across a sum or difference. It's the cornerstone of polynomial multiplication.
FOIL Method: A mnemonic for multiplying two binomials, representing First, Outer, Inner, Last terms. While useful, it's essential to understand the underlying distributive property.
Combining Like Terms: Simplifying an expression by adding or subtracting terms with the same variable and exponent.
Exponents: Understanding the rules of exponents, such as xᵃ * xᵇ = xᵃ⁺ᵇ, is crucial for polynomial multiplication.

By mastering these concepts and techniques, you can confidently tackle a wide range of algebraic expressions and simplify them to their most concise forms.

Applications and Importance

Understanding polynomial multiplication and simplification is not just an academic exercise; it has practical applications in various fields, including:

Engineering: Designing structures, circuits, and systems often involves working with polynomial equations.
Physics: Modeling physical phenomena, such as projectile motion or wave behavior, frequently requires polynomial manipulation.
Computer Graphics: Creating realistic images and animations relies on polynomial functions to describe curves and surfaces.
Economics: Modeling economic trends and forecasting market behavior can involve polynomial equations.

Furthermore, these skills are essential for advanced mathematical studies, such as calculus and differential equations. A solid foundation in polynomial manipulation is crucial for success in these areas.

In conclusion, expanding and simplifying the expression (7x²) (2x³+5) (x²-4x-9) demonstrates the power of algebraic techniques and their importance in various fields. By systematically applying the distributive property, combining like terms, and understanding the rules of exponents, we can unravel complex expressions and reveal their underlying structure. This ability is not only a valuable mathematical skill but also a crucial tool for problem-solving in a wide range of disciplines.

Polynomial multiplication is a fundamental skill in algebra, serving as a building block for more advanced mathematical concepts. One expression that exemplifies the process of polynomial multiplication is (7x²) (2x³+5) (x²-4x-9). This article provides a comprehensive guide to expanding and simplifying this expression, offering a step-by-step approach that demystifies the process. We will delve into the underlying principles, techniques, and practical applications of polynomial multiplication, empowering you to tackle similar expressions with confidence.

Unpacking the Components: A Detailed Look at (7x²) (2x³+5) (x²-4x-9)

The expression (7x²) (2x³+5) (x²-4x-9) is composed of three factors: a monomial (7x²), a binomial (2x³+5), and a trinomial (x²-4x-9). Our goal is to multiply these factors together to obtain a single polynomial expression in its simplest form. This involves applying the distributive property, combining like terms, and adhering to the rules of exponents. The process may seem daunting at first, but by breaking it down into manageable steps, we can conquer this algebraic challenge.

Before we embark on the multiplication journey, let's define some key terms:

Monomial: An algebraic expression consisting of one term, such as 7x².
Binomial: An algebraic expression consisting of two terms, such as (2x³+5).
Trinomial: An algebraic expression consisting of three terms, such as (x²-4x-9).
Polynomial: An algebraic expression consisting of one or more terms, each of which is a product of a constant and one or more variables raised to non-negative integer powers.

With these definitions in mind, we can proceed with the multiplication process.

Step 1: Strategic Grouping: Multiplying the Binomial and Trinomial First

When multiplying multiple factors, we have the flexibility to choose the order in which we perform the multiplications. In this case, a strategic approach is to first multiply the binomial (2x³+5) and the trinomial (x²-4x-9). This will result in a polynomial with potentially more terms, which we can then multiply by the monomial 7x² in the next step. This approach helps to manage the complexity of the expression and reduces the chances of making errors.

To multiply the binomial and trinomial, we will apply the distributive property. Each term in the binomial must be multiplied by each term in the trinomial. This can be visualized as distributing the binomial across the trinomial, or vice versa:

(2x³+5) (x²-4x-9) = 2x³(x²-4x-9) + 5(x²-4x-9)

Now, we distribute 2x³ and 5 across the terms in the trinomial:

= 2x³ * x² + 2x³ * (-4x) + 2x³ * (-9) + 5 * x² + 5 * (-4x) + 5 * (-9)

Next, we perform the individual multiplications, remembering the rule for multiplying exponents: xᵃ * xᵇ = xᵃ⁺ᵇ:

= 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45

So, the product of the binomial and trinomial is 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45. This is a polynomial with six terms, ranging from degree 5 to degree 0 (the constant term).

Step 2: Unleashing the Monomial: Multiplying by 7x²

Now that we have the product of the binomial and trinomial, 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45, we are ready to multiply this polynomial by the monomial factor, 7x². Again, we apply the distributive property, multiplying each term in the polynomial by 7x²:

7x² (2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45) = 7x² * 2x⁵ + 7x² * (-8x⁴) + 7x² * (-18x³) + 7x² * 5x² + 7x² * (-20x) + 7x² * (-45)

Performing the individual multiplications, we get:

= 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

This is the expanded form of the original expression. It is a polynomial of degree 7, with terms ranging from x⁷ down to x².

Step 3: The Final Flourish: Simplifying the Result

The final step is to simplify the expanded expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we examine the polynomial:

14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

We observe that there are no like terms. Each term has a unique power of x, so no terms can be combined. Therefore, the polynomial is already in its simplest form.

Thus, the final simplified expression is:

(7x²) (2x³+5) (x²-4x-9) = 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

We have successfully expanded and simplified the given expression, arriving at a single polynomial in its most concise form.

Core Principles and Techniques: A Recap of Polynomial Multiplication

Throughout this process, we have utilized several fundamental principles and techniques of polynomial multiplication. Let's summarize them:

Distributive Property: This property is the cornerstone of polynomial multiplication. It allows us to multiply a factor across a sum or difference, ensuring that each term in one factor is multiplied by each term in the other factor.
Order of Operations: When multiplying multiple factors, the order in which we perform the multiplications does not affect the final result (due to the commutative property of multiplication). However, a strategic order can simplify the process.
Multiplying Exponents: When multiplying terms with the same base, we add the exponents: xᵃ * xᵇ = xᵃ⁺ᵇ.
Combining Like Terms: Simplifying an expression by adding or subtracting terms with the same variable and exponent.

Mastering these principles and techniques is crucial for success in algebra and beyond.

The Broader Picture: Applications and Significance of Polynomial Multiplication

Polynomial multiplication is not just an abstract mathematical concept; it has numerous practical applications in various fields:

Engineering: Polynomials are used to model various physical phenomena, such as the trajectory of a projectile or the behavior of electrical circuits. Polynomial multiplication is essential for analyzing and designing such systems.
Computer Science: Polynomials are used in computer graphics to represent curves and surfaces, and polynomial multiplication is used in rendering and animation algorithms.
Economics: Polynomials can be used to model economic relationships, such as cost curves and revenue functions. Polynomial multiplication can be used to analyze these models and make predictions.
Statistics: Polynomials are used in statistical modeling, such as regression analysis. Polynomial multiplication is used in fitting models to data.

Moreover, polynomial multiplication is a foundational skill for more advanced mathematical topics, such as calculus, differential equations, and linear algebra. A strong understanding of polynomial multiplication is essential for success in these areas.

In conclusion, expanding and simplifying the expression (7x²) (2x³+5) (x²-4x-9) provides a valuable illustration of the process of polynomial multiplication. By applying the distributive property, combining like terms, and adhering to the rules of exponents, we can transform a complex expression into a simpler, more manageable form. This skill is not only crucial for success in mathematics but also has broad applications in various scientific, engineering, and economic disciplines. Mastering polynomial multiplication empowers us to solve problems, model phenomena, and make informed decisions in a wide range of contexts.

Algebraic expressions can often appear daunting, with their mix of variables, coefficients, and exponents. However, by employing a systematic approach and understanding the fundamental principles of algebra, we can unravel their complexities and arrive at simplified solutions. One such expression that exemplifies this process is (7x²) (2x³+5) (x²-4x-9). This article provides a detailed, step-by-step solution for expanding and simplifying this expression, shedding light on the underlying algebraic techniques and concepts involved. We will break down the expression into manageable parts, apply the distributive property, combine like terms, and ultimately arrive at a concise and meaningful result.

Deconstructing the Expression: Understanding the Components of (7x²) (2x³+5) (x²-4x-9)

The expression (7x²) (2x³+5) (x²-4x-9) is a product of three factors: the monomial 7x², the binomial (2x³+5), and the trinomial (x²-4x-9). To expand and simplify this expression, we need to multiply these factors together in a systematic manner. This involves applying the distributive property, which states that multiplying a sum by a factor is the same as multiplying each term in the sum by the factor. We will also need to combine like terms, which are terms that have the same variable raised to the same power. The goal is to transform the expression into a single polynomial in its simplest form.

Before we begin the multiplication process, let's define the key terms involved:

Monomial: An algebraic expression consisting of a single term, such as 7x².
Binomial: An algebraic expression consisting of two terms, such as (2x³+5).
Trinomial: An algebraic expression consisting of three terms, such as (x²-4x-9).
Polynomial: An algebraic expression consisting of one or more terms, each of which is a product of a constant and one or more variables raised to non-negative integer powers.
Distributive Property: A fundamental property of algebra that states that a(b + c) = ab + ac.
Like Terms: Terms that have the same variable raised to the same power, such as 3x² and -5x².

With these definitions in hand, we can proceed with the step-by-step solution.

Step 1: The Strategic First Step: Multiplying the Binomial and Trinomial

When multiplying multiple factors, we have the freedom to choose the order in which we perform the multiplications. A strategic approach in this case is to first multiply the binomial (2x³+5) and the trinomial (x²-4x-9). This will result in a polynomial with multiple terms, which we can then multiply by the monomial 7x² in the subsequent step. This approach helps to break down the problem into smaller, more manageable steps.

To multiply the binomial and trinomial, we apply the distributive property. Each term in the binomial must be multiplied by each term in the trinomial. This can be visualized as distributing the binomial across the trinomial, or vice versa:

(2x³+5) (x²-4x-9) = 2x³(x²-4x-9) + 5(x²-4x-9)

Now, we distribute 2x³ and 5 across the terms in the trinomial:

= 2x³ * x² + 2x³ * (-4x) + 2x³ * (-9) + 5 * x² + 5 * (-4x) + 5 * (-9)

Next, we perform the individual multiplications, remembering the rule for multiplying exponents: xᵃ * xᵇ = xᵃ⁺ᵇ:

= 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45

Thus, the product of the binomial and trinomial is 2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45. This is a polynomial with six terms, ranging from degree 5 to degree 0 (the constant term).

Step 2: Unleashing the Monomial: Multiplying by 7x²

7x² (2x⁵ - 8x⁴ - 18x³ + 5x² - 20x - 45) = 7x² * 2x⁵ + 7x² * (-8x⁴) + 7x² * (-18x³) + 7x² * 5x² + 7x² * (-20x) + 7x² * (-45)

Performing the individual multiplications, we get:

= 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

This is the expanded form of the original expression. It is a polynomial of degree 7, with terms ranging from x⁷ down to x².

Step 3: The Final Simplification: Combining Like Terms

The final step in the process is to simplify the expanded expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we examine the polynomial:

14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

We observe that there are no like terms. Each term has a unique power of x, so no terms can be combined. Therefore, the polynomial is already in its simplest form.

Thus, the final simplified expression is:

(7x²) (2x³+5) (x²-4x-9) = 14x⁷ - 56x⁶ - 126x⁵ + 35x⁴ - 140x³ - 315x²

We have successfully expanded and simplified the given expression, arriving at a single polynomial in its most concise form.

Core Concepts and Techniques: A Summary of Algebraic Simplification

Throughout this step-by-step solution, we have utilized several core concepts and techniques of algebraic simplification. Let's recap them:

Distributive Property: This property is fundamental to multiplying polynomials. It allows us to distribute a factor across a sum or difference, ensuring that each term is multiplied correctly.
Order of Operations: When multiplying multiple factors, the order in which we perform the multiplications does not affect the final result (due to the commutative property of multiplication). However, a strategic order can simplify the process.
Multiplying Exponents: When multiplying terms with the same base, we add the exponents: xᵃ * xᵇ = xᵃ⁺ᵇ.
Combining Like Terms: Simplifying an expression by adding or subtracting terms with the same variable and exponent.

Mastering these concepts and techniques is essential for success in algebra and higher-level mathematics.

Real-World Relevance: Applications of Algebraic Simplification

Algebraic simplification is not just an abstract exercise; it has numerous practical applications in various fields:

Engineering: Engineers use algebraic simplification to design structures, circuits, and systems. Polynomial equations are often used to model physical phenomena, and simplifying these equations is crucial for analysis and design.
Computer Science: Computer scientists use algebraic simplification in algorithm design and analysis. Polynomials are used to represent data structures and algorithms, and simplifying these expressions can improve efficiency.
Physics: Physicists use algebraic simplification to model physical systems. Equations describing motion, energy, and other physical quantities often involve polynomials, and simplifying these equations is essential for solving problems.
Economics: Economists use algebraic simplification to model economic relationships. Supply and demand curves, cost functions, and other economic models often involve polynomials, and simplifying these expressions can help to analyze market behavior.

In conclusion, expanding and simplifying the expression (7x²) (2x³+5) (x²-4x-9) provides a valuable illustration of the power and importance of algebraic techniques. By applying the distributive property, combining like terms, and adhering to the rules of exponents, we can transform a complex expression into a simpler, more manageable form. This skill is not only crucial for success in mathematics but also has broad applications in various scientific, engineering, and economic disciplines. Mastering algebraic simplification empowers us to solve problems, model phenomena, and make informed decisions in a wide range of contexts.