/*! 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 14 Rewrite the expression by taking... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 14

Rewrite the expression by taking out the common factors. \(9 x^{2}+18 x+3\)

Short Answer

Expert verified
Answer: The factored form of the expression is 3(\(3x^2+6x+1\)).

Step by step solution

01

Identifying the Common Factor

Examine each term in the given expression. The expression is \(9x^2 + 18x + 3\). The common factor for the coefficients of all the terms is 3.
02

Factoring Out the Common Factor

We will now factor out the common factor (3) from each term in the expression. So we have: 3(\(3x^2+6x+1\))
03

Simplifying the Expression (if necessary)

In this case, there is no further simplification needed. So, the final factored expression is: 3(\(3x^2+6x+1\))

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.

Understanding Common Factors
A common factor is a number or variable that evenly divides each term in an algebraic expression. When we look at the expression \(9x^2 + 18x + 3\), we see that each term has a common factor of 3, since 3 divides 9, 18, and 3 evenly. Recognizing common factors is crucial because it allows us to "take out" or factor these elements from all terms in the expression, simplifying it to a simpler form. This helps in easier computations and provides clarity when solving equations. To find a common factor:
  • Examine each term individually.
  • Determine numbers or variables that can be divided evenly from each of the terms.
In our exercise, we identified the number 3 that is common across all coefficients, which is why it's factored out.
Introduction to Polynomials
Polynomials are mathematical expressions consisting of variables raised to whole number exponents and coefficients, combined using addition, subtraction, or multiplication. An expression like \(9x^2 + 18x + 3\) is a polynomial, specifically a quadratic polynomial because the highest power of the variable, \(x\), is 2. Polynomials can have:
  • Terms: Each separate part of a polynomial, like \(9x^2\), \(18x\), and 3.
  • Coefficients: Numbers multiplying the variables, such as 9, 18, and 3 in our example.
  • Degrees: The highest power of the variable, which is 2 for \(x^2\) here.
Understanding polynomials is fundamental in algebra since many equations involve them. They can represent everything from simple numbers to more complex mathematical models.
Simplifying Expressions Effectively
Simplifying expressions involves reducing them to their easiest form by performing operations like factoring out common factors. The goal is to rewrite the expression in the simplest way possible. In our example, the expression \(9x^2 + 18x + 3\) simplifies to \(3(3x^2 + 6x + 1)\) after taking out the common factor of 3.Steps to simplify:
  • Identify any common factors, as demonstrated above.
  • Factor those elements out, as we grouped the expression as \(3(3x^2 + 6x + 1)\).
  • Check if the new expression can be reduced further. In this case, \(3(3x^2 + 6x + 1)\) is already simplified since there are no more common factors.
Simplifying expressions makes them easier to interpret and solve, especially within equations.

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

Write each expression as a sum, difference, or product of two or more algebraic fractions. There is more than one correct answer. Assume all variables are positive. $$ \frac{z+1}{2} $$

Multiply and simplify. Assume any factors you cancel are not zero. $$ \frac{w^{2} r+4 w r}{2 r^{2} w+2 w r} \cdot \frac{r+r^{2}}{4 w+16} $$

If \(a-b+c=17\), what is \(2(a+1)-(b+3)+(2 c-b)\) ?

Factor each expression completely. $$ q^{4}-q^{8} $$

Factor each expression completely. $$ (t+1)^{3}-25(t+1) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied 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.