/*! 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 77 Factor each polynomial completel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 77

Factor each polynomial completely. $$(x+3) x+(x+3) 7$$

Short Answer

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

Step by step solution

01

Identify Common Factors

First, observe the polynomial ewline (x+3)x + (x+3)7. Here, (x+3) is a common factor in both terms.
02

Factor Out the Common Factor

Factor out the common factor (x+3) from each term: ewline (x+3)x + (x+3)7 = (x+3)(x+7).
03

Verify the Factorization

To verify, expand the factored expression to see if it matches the original polynomial: ewline (x+3)(x+7) = x(x+7) + 3(x+7). ewline Distribute both x and 3: ewline x^2 + 7x + 3x + 21. ewline Combine like terms: ewline x^2 + 10x + 21, ewline which matches the original polynomial. This confirms that the factorization is correct.

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.

Common Factors
Before diving into factoring polynomials, it鈥檚 crucial to understand common factors. A common factor is a term that divides each term in the polynomial without leaving any remainder.
In our example polynomial, (x+3)x + (x+3)7, we notice that the term (x+3) appears in both parts of the expression.
This makes (x+3) a common factor.
Identifying common factors is the first step in simplifying polynomials. Once found, you can factor them out from each term, reducing the polynomial to a simpler form.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler, multiplied terms called factors.
Using our example, (x+3)x + (x+3)7, we identified (x+3) as the common factor.
We factored this out, resulting in (x+3)(x+7).
This process simplifies the polynomial and makes it easier to manipulate or solve in equations.
  • Step 1: Identify the common factor, in this case, (x+3).
  • Step 2: Factor out the common factor from each term, giving us (x+3)(x+7).
Polynomial factorization not only simplifies expressions but also helps in solving polynomial equations more efficiently.
Verification of Factorization
After factoring a polynomial, it's essential to verify the factorization to ensure it's correct.
For our example, we factored (x+3)x + (x+3)7 into (x+3)(x+7).
To verify, we need to expand the factored form to see if it matches the original polynomial:
  • Expand (x+3)(x+7) using the distributive property.
  • First, distribute x: x(x+7) = x^2+7x.
  • Next, distribute 3: 3(x+7) = 3x+21.
  • Combine like terms: x^2 + 7x + 3x + 21 = x^2 + 10x + 21.
Since x^2 + 10x + 21 is equivalent to the original expression, our factorization is verified as correct.
Verification is a crucial step to avoid errors and confirm the accuracy of your factored polynomial.

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 each polynomial using the trial-and-error method. $$ 3 x^{2}-17 x-6 $$

Factor each polynomial using the trial-and-error method. $$ a^{2}+7 a+10 $$

Factor each polynomial using the trial-and-error method. $$ 2 w^{2}+21 w-11 $$

Factor each polynomial using the trial-and-error method. $$ z^{2}-2 z-35 $$

Factor each polynomial using the trial-and-error method. $$ x^{2}-8 x+12 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.