Problem 72 Simplify the given expression. ... [FREE SOLUTION]

91影视

Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 72

Simplify the given expression. $$\left(2 x^{2}\right)\left(5 x y-y^{2}\right)-x^{2} y^{2}$$

Short Answer

Expert verified

The simplified expression is \10 x^3 y - 2 x^4 y^2 - x^2 y^2\.

Step by step solution

Distribute the First Expression

Start by distributing the term $(2 x^2)$ to both terms within the parentheses $(5xy - y^2)$.$(2 x^2)(5xy) - (2 x^2)(y^2)$This will give:$(10 x^3 y) - (2 x^4 y^2)$.

Combine Like Terms

Rewrite the expanded terms including the remaining $x^2 y^2$, and simplify by combining like terms:$10 x^3 y - 2 x^4 y^2 - x^2 y^2$.

Identify Like Terms

Recognize that there are no like terms to combine further. The given expression is simplified as is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

The distributive property is a fundamental concept in algebra. It involves distributing one term across terms inside parentheses. To distribute, multiply the term outside the parentheses by each term inside. For example, consider $a(b + c)$. Using the distributive property, we get $ab + ac$.

In the exercise, the term $2x^2$ is distributed to both $5xy$ and $y^2$. The detailed steps are: $2x^2(5xy) - 2x^2(y^2)$. This simplifies to $10x^3y - 2x^4y^2$.

Understanding the distributive property helps in simplifying expressions, solving equations, and expanding polynomials.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have identical variable parts raised to the same power. For instance, $3x^2$ and $7x^2$ are like terms, but $3x^2$ and $3x$ are not.

After distributing in our exercise, we get $10x^3y - 2x^4y^2 - x^2y^2$. It's clear that there are no terms with the same variable parts and exponents. Thus, no further combination is possible.

Understanding how to combine like terms makes it easier to simplify complex expressions and solve equations accurately.

Polynomial Expressions

A polynomial expression consists of variables, constants, and exponents, combined using addition, subtraction, and multiplication. Each part of a polynomial is called a 'term'. For instance, in $3x^2 + 5x - 7$, there are three terms: $3x^2$, $5x$, and $-7$.

In our exercise, $2x^2(5xy - y^2) - x^2y^2$ is a polynomial expression involving multiplication and subtraction. Distribute and simplify by combining terms to achieve $10x^3y - 2x^4y^2 - x^2y^2$.

Polynomials are foundational in algebra, helping to solve a wide range of mathematical problems.

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91影视

Simplify the given expression. $$\left(2 x^{2}\right)\left(5 x y-y^{2}\right)-x^{2} y^{2}$$

Short Answer

Step by step solution

Distribute the First Expression

Combine Like Terms

Identify Like Terms

Key Concepts

Distributive Property

Combining Like Terms

Polynomial Expressions

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