/*! 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 22 Factor each trinomial, or state ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 22

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+4 x+5$$

Short Answer

Expert verified
The trinomial \(x^{2}+4 x+5\) is prime in the integers, but it can be factored in the complex numbers as \((x-(-2+i))(x-(-2-i))\).

Step by step solution

01

Identifying the relevant terms

Identifying the relevant terms of trinomial \(x^{2}+4 x+5\). Here, a equals to 1 (coefficient of \(x^{2}\)), b equals to 4 (coefficient of x), and c equals to 5 (the constant term).
02

Factoring the Trinomial

Start factoring the trinomial. Look for two numbers that multiply to \(a*c=(1*5)=5\) and add up to b=4. However, there are no such numbers, meaning the quadratic trinomial cannot be factored by traditional factoring. The trinomial is prime in the integers but not in the complex numbers.
03

The Complex Factorization

A prime trinomial in the integers can still often be factored in the complex numbers. Use the quadratic formula \(x = [-b \pm sqrt(b^{2}-4ac)]/(2a)\). Substitute a=1, b=4, c=5. Calculating, we get two complex roots \(x = -2+ i\) and \(x = -2- i\). Hence, trinomial in the complex numbers can be factored as \(x^{2}+4 x+5=(x-(-2+i))(x-(-2-i))\).
04

Verifying with FOIL Multiplication

To verify the factorization, FOIL (First, Outer, Inner, Last) multiplication is applied to distribute the binomials. First, multiply the first terms of each binomial. Then, multiply the outer terms, followed by the inner terms. Last, multiply the last terms of each binomial. Add up all the products to verify if it equals to the original trinomial.

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

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$4 a y^{3}-12 a y^{2}+9 a y$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$24 a^{4} b+60 a^{3} b^{2}+150 a^{2} b^{3}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 x^{4}+6 x^{3} y+2 x^{2} y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$81 x^{4} y-y^{5}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x^{2}-4 x y-12 y^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.