/*! 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 40 Factor completely. If a polynomi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 40

Factor completely. If a polynomial is prime, state this. $$ 50 t^{3}+115 t^{2}+60 t $$

Short Answer

Expert verified
(a^{3} + b)[5c - 1]

Step by step solution

01

Identify Common Factors

Look for common factors in the given expression: o The expression is: 5c(a^{3} + b) - (a^{3} + b) Find the common factor between the two terms, which is (a^{3} + b).
02

Factor Out the Common Factor

Factor out the common factor (a^{3} + b) from each term:(a^{3} + b) [5c - 1]
03

Verify the Factored Form

Check to ensure that factoring out (a^{3} + b) from the original expression gives the factored form: (a^{3} + b)[5c - 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.

Common Factors
Understanding common factors is key to solving polynomial problems. A common factor is a term that divides all terms in the expression evenly. For example, in the expression \(5c(a^{3} + b) - (a^{3} + b)\), we notice that \(a^{3} + b\) is present in both terms. This term is the common factor. Think of common factors like common ingredients in a recipe. By identifying these ingredients, we can simplify the recipe. In math, finding common factors helps us simplify the equation.
To find common factors:
  • List the factors of each term.
  • Identify the factors that appear in every term.
  • The greatest common factor is the largest factor common to all terms.
Once identified, you can factor it out to simplify the polynomial.
Factoring Out
Factoring out is the process of taking the common factor from each term and placing it outside a set of parentheses. In the expression \(5c(a^{3} + b) - (a^{3} + b)\), we factor out \(a^{3} + b\). This step transforms the expression into \((a^{3} + b) \left(5c - 1\right)\). Here鈥檚 a step-by-step guide to factoring out:
  • Identify the common factor in all terms.
  • Rewrite each term as the product of the common factor and the remaining factor.
  • Place the common factor outside the parentheses.
  • Combine the terms inside the parentheses.
Factoring out simplifies the original equation and makes it easier to handle. It is like pulling a common ingredient out of a mix to see what鈥檚 left.
Polynomial Prime Check
Not all polynomials can be factored. A polynomial is considered 'prime' if it has no factors other than 1 and itself. To check if a polynomial is prime:
  • Verify if there are any common factors.
  • Attempt to factor out any possible terms.
  • If no common factors or groups exist, the polynomial is prime.
For our example, after factoring out, we have \((a^{3} + b) [5c - 1]\). Neither of these factors can be reduced further, indicating that the polynomial is completely factored. If no common factor was found, the polynomial would be prime. Identifying prime polynomials tells us that the expression is in its simplest form.

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

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}-3 x-7=0 $$

Akio concludes that since \(x^{2}-9=(x-3)(x+3)\) it must follow that \(x^{2}+9=(x+3)(x-3) .\) What mistake(s) is he making?

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{-6}+x^{-9}+x^{-3} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ 9 x^{2 n}-6 x^{n}+1 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}-16 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.