/*! 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 42 Factor each polynomial by groupi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 42

Factor each polynomial by grouping. $$2 w t-2 w a+t-a$$

Short Answer

Expert verified
(t - a)(2w + 1)

Step by step solution

01

Group the Terms

Group the polynomial terms into two pairs. The given polynomial is: \[ 2wt - 2wa + t - a \] Group them as: \[ (2wt - 2wa) + (t - a) \]
02

Factor out the Greatest Common Factor (GCF) from Each Group

Factor out the GCF from each group of terms. The first group is \[2wt - 2wa\], the GCF is \[2w \]. The second group is \[t - a\], no GCF other than 1, so it remains the same. This gives: \[ 2w(t - a) + (t - a) \]
03

Factor Out the Common Binomial Factor

Now, factor out the common binomial factor \[ (t - a) \] from both terms: \[ 2w(t - a) + (t - a) = (t - a)(2w + 1) \]

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.

Grouping Method
The grouping method is a useful technique for factoring polynomials, especially when dealing with four-term polynomials. This technique involves grouping the terms of the polynomial into smaller sets, making it easier to identify common factors.
To use the grouping method correctly:
  • First, pair the terms together in a way that makes factoring simpler. For example, with the polynomial \[ 2wt - 2wa + t - a \], group them as \[ (2wt - 2wa) + (t - a) \].
  • Then, factor out any common factors within these groups.
  • Once the groups are factored, look for a common binomial factor.
By breaking the polynomial into smaller groups, you can simplify complex polynomial expressions step by step.
Greatest Common Factor
The Greatest Common Factor (GCF) is essential for factoring polynomials. It's the highest factor that divides all terms in a polynomial.
In the process of factoring, always begin by finding the GCF for each group of terms.
For example, in the polynomial \[ 2wt - 2wa + t - a \], we have two groups: \[ 2wt - 2wa \] and \[ t - a \].
  • For the first group \[2wt - 2wa\], the GCF is \[ 2w \]. Factoring this out, we get \[ 2w(t - a) \].
  • For the second group \[t - a\], the GCF is 1, so no change is necessary.
Thus, factoring out the GCF helps in simplifying and streamlining the polynomial, which is crucial for the next steps in factoring.
Binomial Factorization
Binomial factorization involves identifying and factoring out common binomial expressions from a polynomial.
This is the final step after grouping and factoring out the GCF.
Consider the polynomial from our exercise: \[ 2w(t - a) + (t - a) \]. Here, \[ (t - a) \] is the common binomial factor in both terms.
To factor out \[ (t - a) \], rewrite the expression as:
  • \[ (t - a)(2w + 1) \]
This step simplifies the polynomial further by combining the factored parts into a product of binomials, making it much easier to handle and understand.
Remember, identifying the common binomial factor is crucial for accurate and effective polynomial factorization.

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

Work with a group to examine the following steps in the solution to \(5 x^{2}-5=0\) $$\begin{array}{r}5\left(x^{2}-1\right)=0 \\\5(x-1)(x+1)=0 \\\x-1=0 \quad \text { or } \quad x+1=0 \\\x=1 \quad \text { or } \quad x=-1\end{array}$$ What happened to the 5? Explain.

Solve each equation. $$16 x^{2}+8 x+1=0$$

Factor each polynomial by grouping. $$ w b+3 w+12+4 b $$

Tossing a ball. A boy tosses a ball upward at 32 feet per second from a window that is 48 feet above the ground. The height of the ball above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+32 t+48$$ Find the time at which the ball strikes the ground.

Factor each polynomial using the trial-and-error method. $$ 7 a^{2}-8 a+1 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.