Problem 83 Factor. $$a\left(x^{2}-4\right... [FREE SOLUTION]

Chapter 10: Problem 83

Factor. $$a\left(x^{2}-4\right)+b\left(x^{2}-4\right)$$

Short Answer

Expert verified

The factored form of the given expression is $(x^{2}-4)(a+b)$

Step by step solution

- Identification of common factors

The first step in solving the exercise is recognizing that both terms of the expression $a(x^{2}-4)+b(x^{2}-4)$ share the same common factor, which is $(x^{2}-4)$.

- Applying the distributive property

In the second step, we apply the distributive property (also known as the factorization method) to take out the common factor. This method says that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the products. Here, it gives us: $(x^{2}-4)(a+b)$.

- Final Answer

Hence, the factored form of the provided expression is $(x^{2}-4)(a+b)$.

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Key Concepts

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

Common Factor

When we talk about a common factor, we mean a term that is present in multiple parts of an expression. Recognizing a common factor is a crucial step in many algebra problems. It helps simplify expressions and solve equations more efficiently.
For example, in the expression $a(x^{2}-4)+b(x^{2}-4)$, the term $(x^{2}-4)$ appears in both parts of the expression. This indicates that $(x^{2}-4)$ is a common factor.
Finding the common factor is like finding the greatest common divisor (GCD) in arithmetic. We extract this common part to simplify problems, making them easier to handle. Keep an eye out for repeated terms since they hint at a possible common factor.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions in a systematic way. It states that describing how a sum or difference of terms can be multiplied by a common factor.
In the expression $a(x^{2}-4) + b(x^{2}-4)$, once we've identified $(x^{2}-4)$ as a common factor, the distributive property lets us "factor it out."
This means we can rewrite the expression as $(x^{2}-4)(a+b)$. We're essentially distributing the common factor over the terms inside the parentheses. By doing this, we make the expression more compact and manageable.

Factorization

Factorization is the process of breaking down an expression into the product of its factors. It's a powerful tool that can transform complex expressions into more digestible parts. This is particularly useful for solving equations and simplifying expressions.
By applying the distributive property to the expression $a(x^{2}-4) + b(x^{2}-4)$, we have successfully factorized it into $(x^{2}-4)(a+b)$.
This factorization shows each distinct part clearly, making it easier to understand and work with.

Algebra

Algebra is all about finding ways to solve equations and understand relationships between quantities. It's about looking for patterns, making generalizations, and solving for unknowns.
In this exercise, involving the expression $a(x^{2}-4) + b(x^{2}-4)$, we used key algebraic strategies: finding common factors and applying the distributive property to simplify it.
Through algebraic thinking, we learn to manipulate symbols and create a more profound understanding of numbers and equations. By mastering algebra, you gain a toolkit for solving various mathematical challenges efficiently and creatively.

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91影视

Factor. $$a\left(x^{2}-4\right)+b\left(x^{2}-4\right)$$

Short Answer

Step by step solution

- Identification of common factors

- Applying the distributive property

- Final Answer

Key Concepts

Common Factor

Distributive Property

Factorization

Algebra

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