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Chapter 6: Problem 75

Write an equivalent expression by factoring. $$ (x+7)(x-1)+(x+7)(x-2) $$

Short Answer

Expert verified
(x+7)(2x-3)

Step by step solution

01

- Identify the common factor

Observe that \(x+7\) is a common factor in both terms of the expression \( (x+7)(x-1) + (x+7)(x-2) \).
02

- Factor out the common factor

Since \(x+7\) is common, factor it out: \( (x+7)((x-1) + (x-2)) \).
03

- Simplify inside the parentheses

Combine the terms inside the parentheses: \( (x-1) + (x-2) = 2x - 3 \). So the expression becomes \ (x+7)(2x-3) \.

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Key Concepts

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

Common Factor
When working with algebraic expressions, identifying a common factor is essential. A common factor is a term or expression that appears in each part of an expression.
In the expression given, \((x+7)(x-1)+(x+7)(x-2)\), notice that \((x+7)\) appears in both terms. Recognizing this common factor allows us to simplify the expression effectively.
Always look for common terms or expressions when first examining an algebraic problem. This step can often lead to a simpler and more manageable form of the expression.
Factoring Out
Factoring out involves taking the common factor outside of the expression, simplifying what remains. After identifying the common factor \((x+7)\) in the expression \((x+7)(x-1)+(x+7)(x-2)\), we rewrite the expression as:
\( (x+7)((x-1) + (x-2)) \)
This process reduces the initial complex expression into a simpler form. The goal is to make the expression easier to work with by isolating the common term.
Factoring out is a powerful tool in algebraic manipulation, making subsequent steps like simplification straightforward.
Simplifying Expressions
Simplifying expressions is the final and often the most rewarding step. Once you've factored out the common factor, you work on the remaining expression within the parentheses:\
\( (x-1) + (x-2) = 2x - 3 \)
Combining like terms (in this case, the \(x\)'s and the constants), we get the simplified form. The initial expression \((x+7)(x-1)+(x+7)(x-2)\) becomes:
\((x+7)(2x-3)\)
Simplification often reduces an expression to its most basic form, making it easier to understand and work with. Always double-check each step to ensure accuracy and clarity in your final result.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 a^{4}-25 a^{2} b^{4} $$

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Akio concludes that since \(x^{2}-9=(x-3)(x+3)\) it must follow that \(x^{2}+9=(x+3)(x-3) .\) What mistake(s) is he making?

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