/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Write an equivalent expression b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 45

Write an equivalent expression by factoring out the greatest common factor. $$ 2 t^{2}+8 t $$

Short Answer

Expert verified
Factor out the GCF to get \(2(t^2 + 4t)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the expression. The terms in our expression are \(2t^2\) and \(8t\). The GCF of 2 and 8 is 2.
02

Factor Out the GCF

Next, factor out the GCF from each term in the expression. Divide each term by 2 and write the expression as a product of the GCF and the remaining factors. \[2t^2 + 8t = 2(t^2) + 2(4t) = 2(t^2 + 4t)\]
03

Verify the Factored Expression

To verify the factored expression, multiply the GCF by each term inside the parentheses and check if it simplifies back to the original expression. \[2(t^2 + 4t) = 2t^2 + 8t\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Greatest Common Factor (GCF)
To solve an algebraic expression by factoring, we first need to understand the greatest common factor, or GCF. The GCF is the largest number that can divide all terms of an expression without leaving a remainder. This helps simplify the expression by reducing it to a more manageable form. For example, in the expression \(2t^2 + 8t\), the terms are \(2t^2\) and \(8t\). Identify the GCF by looking at the coefficients and the variables. Here, the highest number that can divide both 2 and 8 is 2. We also notice both terms have at least one 't'. However, in factoring out the GCF, we use only the numerical factor, which is 2. So, the GCF is just 2.
Polynomial Factoring
Factoring polynomials involves expressing the polynomial as a product of simpler polynomials. Once you've identified the GCF as explained, the next step is to factor it out from each term. In our example, the initial expression is \(2t^2 + 8t\). Since the GCF is 2, you factor out 2 from each term. This gives us the following:

\[2t^2 + 8t = 2(t^2) + 2(4t) = 2(t^2 + 4t)\]

This step helps simplify the polynomial into a product involving the GCF and another polynomial \(t^2 + 4t\), which is easier to work with for further operations.
Algebraic Simplification
Algebraic simplification is the process of reducing complex expressions to their simplest form. After factoring out the GCF, it鈥檚 crucial to verify the expression is correct. Simplification verifies that the factored form returns to the original expression when multiplied back.

Let's check:
  • Take the factored expression: \(2(t^2 + 4t)\)
  • Distribute the 2 back into each term inside the parentheses: \[2(t^2) + 2(4t) = 2t^2 + 8t\]
This confirms that the factored expression is equivalent to the original. Simplification is key to solving and verifying algebraic expressions efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor completely. If a polynomial is prime, state this. $$ 64 t^{6}-1 $$

Solve using the principle of zero products. $$ (x+3)(x-4)=0 $$

A silo is a structure that is shaped like a right circular cylinder with a half sphere on top. The surface area of a silo of height \(h\) and radius \(r\) (including the area of the base) is given by the polynomial \(2 \pi r h+\pi r^{2} .\) (Note that \(h\) is the height of the entire silo.) Find an equivalent expression by factoring out a common factor. (IMAGE CANT COPY)

Factor completely. If a polynomial is prime, state this. $$ 3 b^{2}+17 a b-6 a^{2} $$

Factor completely. $$ 64-125 x^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.