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Chapter 5: Problem 98

Factor each expression. $$(3 t+5)^{2}-(3 t+5)$$

Short Answer

Expert verified
The factored expression of \((3 t+5)^{2}-(3 t+5)\) is \((3t+5)(3t+4)\)

Step by step solution

01

Identify the common factor

In the given expression \((3t+5)^2 - (3t+5)\), the common factor is \(3t + 5\)
02

Factor out the common factor

The next step is to take this common factor out from both terms. So, \((3t+5)^2 - (3t+5) = (3t+5)\[(3t+5) - 1\]\)
03

Simplify the expression

Now, simplify the expression inside the brackets. So, \((3t+5)\[(3t+5) - 1\] = (3t+5)(3t+4)\)

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Key Concepts

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

Factoring Expressions
Factoring expressions is a core skill in algebra that allows you to simplify expressions by breaking them down into simpler parts. This process involves expressing a complicated expression as a product of its factors. In the example given, we dealt with the expression \((3t+5)^2 - (3t+5)\). Here, our goal is to rewrite this expression in a simpler, more manageable form.

By factoring, we transform expressions to make solving equations easier and to reveal deeper insight into the mathematical relationships they describe. Factoring can sometimes expose potential solutions or further simplify subsequent operations.
Common Factor
A common factor is a term that appears in all parts of an expression, allowing it to be factored out to simplify the expression. Identifying this common term is a crucial step in the factoring process. In the provided exercise, the entire expression \((3t+5)^2 - (3t+5)\) has a common factor of \((3t+5)\).

Once identified, this common factor can be extracted from each part of the expression. This leaves a simpler expression that is easier to work with. It's like finding a common thread through different pieces, pulling them together into one straightforward expression.
  • Identify potential common terms.
  • Check each part of the expression for these terms.
  • Factor them out to simplify further.
Algebraic Simplification
Algebraic simplification is the process of reducing an expression to its simplest form. After factoring out the common term, the expression \((3t+5)^2 - (3t+5)\) becomes \((3t+5)\left[(3t+5) - 1\right]\).

In algebraic simplification, the next step is to perform operations inside any brackets or parentheses. In this case, \((3t+5) - 1\) simplifies to \(3t+4\). So, the entire expression becomes \((3t+5)(3t+4)\). This simplified form is often much easier to work with in further equations or operations.
  • Reduce complex expressions to simpler ones.
  • Simplify operations step-by-step.
  • Aim for clarity and ease of use.


Simplification not only streamlines the problem-solving process but also makes complex algorithms and equations much more manageable.

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Most popular questions from this chapter

When we add \(5 x\) and \(7 x\), we combine like terms: \(5 x+7 x=12 x\). Explain how this is related to factoring out a common factor.

Explain how to use the distributive property to check the factoring of a trinomial.

Factor. $$4 x^{2}-4 x+1$$

Factor. $$x^{2}+5 x-24$$

Write the terms of each trinomial in descending powers of one variable. Then factor. $$16-40 a+25 a^{2}$$
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