Problem 55 Factor completely. $$ 2 t^{3... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 6: Problem 55

Factor completely. $$ 2 t^{3}+8 t^{2}+6 t $$

Short Answer

Expert verified

$2t(t + 3)(t + 1)$

Step by step solution

Identify the Greatest Common Factor (GCF)

To factor the polynomial, start by finding the Greatest Common Factor (GCF) of all the terms. Here the terms are: $2t^{3}$, $8t^{2}$, and $6t$. The GCF is 2t.

Factor out the GCF from each term

Now factor $2t$ out of each term in the polynomial: \[2t(t^{2} + 4t + 3)\]

Factor the quadratic expression

Factor the expression inside the parentheses. We need to find two numbers that multiply to 3 (the constant term) and add to 4 (the coefficient of the linear term). Those numbers are 3 and 1, so we can write: \[2t(t + 3)(t + 1)\]

Write the final factored form

Combine everything obtained from the previous steps to write the fully factored polynomial: \[2t(t + 3)(t + 1)\]

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

To begin factoring any polynomial, the first and crucial step is to find the Greatest Common Factor (GCF) of all the terms. The GCF is the largest factor that divides all the terms in the polynomial. In this exercise, the polynomial given is \[2 t^{3}+8 t^{2}+6 t\]. Let's break it down:

The coefficients are 2, 8, and 6.
The variable is t, appearing in each term.

The GCF of the coefficients (2, 8, and 6) is 2.
The smallest power of t in the polynomial is t. Therefore, the overall GCF is \[2t\].
By factoring out \[2t\], we simplify the expression to\[2t(t^{2} + 4t + 3)\].

Quadratic Expressions

Once we factor out the GCF, we are left with a quadratic expression inside the parentheses. In the form:\[t^{2} + 4t + 3\]
A quadratic expression is a polynomial of degree 2, and it generally takes the form \[ax^{2} + bx + c\].
Here, the expression inside the parentheses confirms this form where \[a = 1, b = 4, \text{and} c = 3\].
The goal is to factor this quadratic expression further.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors.
In the quadratic expression \[t^{2} + 4t + 3\], the factorization process requires finding two numbers that both add up to the middle coefficient (4) and multiply to the constant term (3):

Factors of 3 that add up to 4 are 3 and 1.

Hence, we can write \[t^{2} + 4t + 3\] as \[(t + 3)(t + 1)\].
Combining everything together, we obtain the fully factored form of the original polynomial:\[2t(t + 3)(t + 1)\]. Hence, factorization makes computations easier and reveals essential properties of the polynomial.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor completely. $$ 2 t^{3}+8 t^{2}+6 t $$

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor out the GCF from each term

Factor the quadratic expression

Write the final factored form

Key Concepts

Greatest Common Factor

Quadratic Expressions

Polynomial Factorization

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.