Problem 108 Simplify or solve as appropriate... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 108

Simplify or solve as appropriate. $$(a+b)(a-b)-(a+b)(a+b)$$

Short Answer

Expert verified

The simplification of the given expression is $-2b^2 - 2ab$.

Step by step solution

Expand the expressions

Expand both of the expressions using the distributive property. For the expression $(a+b)(a-b)$, it becomes $a^2 - b^2$. For the expression $(a+b)^2$, this can be rewritten as $(a+b)(a+b)$, which becomes $a^2 + 2ab + b^2$. So, the expression becomes $a^2 - b^2 - (a^2 + 2ab + b^2)$

Simplify the expressions

Combine like terms. You do this by subtracting $a^2$, subtracting $2ab$, and subtracting $b^2$ from $a^2 - b^2$. This results in $a^2 - b^2 - a^2 - 2ab - b^2$. When simplified, this gives $-b^2 - 2ab - b^2$.

Final Simplification

The expression simplifies further to $-2b^2 - 2ab$. This is the simplified form of the given expression.

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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 rule in algebra used to simplify expressions. It allows us to multiply a single term by terms inside parentheses. In this exercise, it's applied to both $(a + b)(a - b)$ and $(a + b)^2$.

For $(a+b)(a-b)$, the distributive property tells us to multiply each term in the first parenthesis by each term in the second, resulting in $a^2 + ab - ab - b^2$, simplifying to $a^2 - b^2$.
For $(a+b)^2$, seen as $(a+b)(a+b)$, each term in the first parenthesis multiplies every term in the second. This generates $a^2 + ab + ab + b^2$, which simplifies to $a^2 + 2ab + b^2$.

Understanding the distributive property is key in breaking down expressions like these, allowing us to transform complex-looking polynomials into more manageable components.

Combining like terms

Combining like terms is a method used to simplify algebraic expressions by merging terms that contain the same variable raised to the same power. After using the distributive property in the example, the expression became $a^2 - b^2 - (a^2 + 2ab + b^2)$.

First, notice the term $a^2$ appears twice, once as positive in $a^2 - b^2$ and once as negative in $a^2 + 2ab + b^2$. When combined, these cancel each other out.
Similarly, $b^2$ is present, which when subtracted becomes $-b^2 - b^2$, simplifying to $-2b^2$.
The term $2ab$ doesn't have a counterpart with the opposite sign, so we simply bring it down as $-2ab$.

Combining these terms yields the final simplified form $-2b^2 - 2ab$. This step is essential for simplifying polynomials to their most concise forms.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured in terms that are powered, added, or subtracted. In solving this problem, we're dealing with polynomial expressions such as $(a+b)(a-b)$ and $(a+b)(a+b)$.

A polynomial like $a^2 - b^2$ is a binomial because it consists of two terms.
The expression $a^2 + 2ab + b^2$ is a trinomial with three terms combining different powers of $a$ and $b$.
Recognizing the type of polynomial helps us apply the correct simplifying techniques, like using the distributive property and combining like terms, to reach the simplified form $-2b^2 - 2ab$.

Polynomials form the backbone of many algebra problems, and mastering their manipulation is essential for tackling more advanced math problems efficiently.

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91影视

Simplify or solve as appropriate. $$(a+b)(a-b)-(a+b)(a+b)$$

Short Answer

Step by step solution

Expand the expressions

Simplify the expressions

Final Simplification

Key Concepts

Distributive property

Combining like terms

Polynomials

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