/*! 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 72 Factor completely, or state that... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 72

Factor completely, or state that the polynomial is prime. $$x^{3}+3 x^{2}-25 x-75$$

Short Answer

Expert verified
The polynomial \(x^{3} + 3x^{2} - 25x -75 \) factors completely as \((x + 5)(x - 5)(x + 3)\).

Step by step solution

01

Identify Potential Rational Roots

Use the Rational Root Theorem to identify potential rational roots of the polynomial. According to the theorem, any possible rational root for our polynomial, \(x^{3}+3 x^{2}-25 x-75\), will be a factor of the constant term (75) divided by a factor of the leading-coefficient (1). Checking systematically, factors of 75 are 卤1, 卤3, 卤5, 卤15, 卤25, 卤75.
02

Synthetic Division

Make a synthetic division by each possible rational root and determine whether it鈥檚 a root or not. This requires dividing the polynomial by (x - p) where 'p' is a potential root. When you get zero at the end of synthetic division, the value you tested is a root of the polynomial. Here, it's found out that -5 is a root.
03

Factoring Further

Since -5 is a root, the factor is (x + 5). Perform synthetic division by this root will reduce the polynomial degree to 2. The resulting quadratic polynomial is \(x^{2} - 2x -15 \). This can be factored further into (x - 5)(x + 3).
04

Final Factored Form

Combine all the factors obtained in previous steps for a final answer. Our polynomial \(x^{3} + 3x^{2} - 25x -75 \) factors to \((x + 5)(x - 5)(x + 3)\).

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!

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division $$\frac{7 x}{x+3}+\frac{(x+3)^{2}}{x-5}$$ I began by dividing the numerator and the denominator by the common factor, \(x+3\).

Rationalize the numerator. $$\frac{\sqrt{x}-\sqrt{y}}{x^{2}-y^{2}}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.