/*! 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 87 Factor the expression by groupin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 87

Factor the expression by grouping terms. $$5 x^{3}+x^{2}+5 x+1$$

Short Answer

Expert verified
The expression factors to \((x^2 + 1)(5x + 1)\).

Step by step solution

01

Recognize the structure

The expression given is \(5x^3 + x^2 + 5x + 1\), which has four terms. To factor by grouping, we first identify two pairs of terms that we can factor separately.
02

Group the first two terms and the last two terms

Group the expression into two pairs: \((5x^3 + x^2)\) and \((5x + 1)\). Now, we'll factor each group separately.
03

Factor out the greatest common factor from each group

In the first group \((5x^3 + x^2)\), we can factor out \(x^2\), which gives \(x^2(5x + 1)\). In the second group \((5x + 1)\), the expression is already in its simplest form, so it remains \(1(5x + 1)\) or simply \(5x + 1\).
04

Combine the common factors

Notice that both groups have \((5x + 1)\) as a factor. So, we can write the expression as \(x^2(5x + 1) + 1(5x + 1)\), which can be factored to \((x^2 + 1)(5x + 1)\) by factoring out \((5x + 1)\).
05

Verify the factored expression

Multiply \((x^2 + 1)(5x + 1)\) to ensure it expands back to the original expression. We find: \((x^2 + 1)(5x + 1) = x^2 \cdot 5x + x^2 \cdot 1 + 1 \cdot 5x + 1 \times 1 = 5x^3 + x^2 + 5x + 1\). The factored form 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.

Polynomial Factoring
Factoring polynomials is a critical concept that transforms complicated expressions into simpler factors. The primary goal is to express a polynomial as a product of simpler polynomials. This can unravel complex equations, making them more manageable and solvable. In our exercise, we started with the polynomial expression \(5x^3 + x^2 + 5x + 1\). Since there are four terms, it suggested the ideal method of factoring by grouping to simplify.

To apply polynomial factoring by grouping, follow these steps:
  • Identify pairs of terms within the polynomial that can be grouped together. These pairs are selected so common factors are present within those groups.
  • Each group of terms is factored separately, revealing common factors.
  • Finally, the overall expression is rewritten using these common factors as a product of polynomials.
Through this systematic approach, polynomial factoring helps in simplifying or even solving equations where direct methods may fail.
Greatest Common Factor
The greatest common factor (GCF) simplifies portions of an equation by identifying the largest factor shared by terms. This factor is then pulled out, making the expression less complex. In the context of our expression \(5x^3 + x^2 + 5x + 1\), determining the GCF is central during the grouping method.

Here's how you use the GCF in factoring by grouping:
  • For the first group \((5x^3 + x^2)\), the GCF is \(x^2\). Factoring \(x^2\) out simplifies it to \(x^2(5x + 1)\).
  • The second group \((5x + 1)\) cannot be factored further, implying the GCF remains \(1\). The expression stays \(5x + 1\).
After factoring out the GCF from these groups, you'll notice a common binomial factor of \((5x + 1)\), crucial for further simplifying the expression. Identifying and using the GCF streamlines the expression considerably, allowing us to reframe it into factors, easing the path for verification.
Verification of Factoring
Verification ensures that the factored expression correctly represents the original equation. It is a crucial step to confirm the accuracy of your factoring process. Once you've factored the polynomial \(5x^3 + x^2 + 5x + 1\) into \((x^2 + 1)(5x + 1)\), verifying this factored form is essential.

Verification involves:
  • Expanding the factored expression \((x^2 + 1)(5x + 1)\).
  • Use basic distributive properties to expand: \((x^2 \cdot 5x) + (x^2 \cdot 1) + (1 \cdot 5x) + (1 \cdot 1)\).
  • Simplify the resulting terms: \(5x^3 + x^2 + 5x + 1\).
If the expanded form matches the original polynomial, which, in our case, it does, the factorization is verified as correct. This step reassures the integrity and correctness of your solution, ensuring no calculation errors were made during the factoring process.

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

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{1.295643 \times 10^{9}}{\left(3.610 \times 10^{-17}\right)\left(2.511 \times 10^{6}\right)}$$

Shrinkage in Concrete Beams As concrete dries, it shrinks - the higher the water content, the greater the shrinkage. If a concrete beam has a water content of \(\bar{w} \mathrm{kg} / \mathrm{m}^{3},\) then it will shrink by a factor $$S=\frac{0.032 w-2.5}{10,000}$$ where \(S\) is the fraction of the original beam length that disappears due to shrinkage. (a) A beam \(12.025 \mathrm{m}\) long is cast in concrete that contains \(250 \mathrm{kg} / \mathrm{m}^{3}\) water. What is the shrinkage factor \(S ?\) How Iong will the beam be when it has dried? (b) A beam is \(10.014 \mathrm{m}\) long when wet. We want it to shrink to \(10.009 \mathrm{m},\) so the shrinkage factor should be \(S=0.00050 .\) What water content will provide this amount of shrinkage? PICTURE CANT COPY

Area of a Region Sketch the region in the coordinate plane that satisfies both the inequalities \(x^{2}+y^{2} \leq 9\) and \(y \geq|x| .\) What is the area of this region?

Scientific Notation Write each number in scientific notation. (a) \(129,540,000\) (b) \(7,259,000,000\) (c) 0.0000000014 (d) 0.0007029

DISCUSS: Proof That \(0=1 ?\) The following steps appear to give equivalent equations, which seem to prove that \(1=0 .\) Find the error. \(\begin{aligned} x &=1 & & \text { Given } \\ x^{2} &=x & & \text { Multiply by } x \\ x^{2}-x &=0 & & \text { Subtract } x \end{aligned}$$(x-1)=0\)Factor \(\frac{(x-1)}{x-1}=\frac{0}{x-1} \quad\) Divide by \(x-1$$x=0 \quad\) Simplify\(1=0 \quad\) Given \(x=1\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

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.