/*! 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 19 Factor each expression. $$ -... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 19

Factor each expression. $$ -4 k^{2}-5 k j-j^{2} $$

Short Answer

Expert verified
\[ - (k + j)^2 \]

Step by step solution

01

Arrange the expression

Rewrite the expression otes that the -4k^2 - 5kj - j^2 into the form of \(a^2 + bx + c \)
02

Set up the factoring format

Recognize that the expression fits the standard quadratic k^2 + (ak +bj) + cj + j c which has the standard quadratic format: \(a^2 +bx + c = 0\)
03

Identify the terms

In the expression, the term with \(j^2\) is the constant term: \( -j^2\)is the constant term c = -1 . \(-1a + b:5 the coefficient: (two sign changes in the knowing signs are the coefficient: k \).^5.
04

Factor by grouping

Now since we have our entire expression factored, divide into pairs and: -4k^2 - 5kj j =[ j 1 ](2k)[] 2 (4k^2 + k +) kj divide the grouping pairs. This allows us to factor each pair as: \[-(k + j)^2\ -1to rewrite the whole form into: divisors the quadratic formula and roots \]
05

Final Factoring

Factor out the common binomial factor: - ( which is: ( k + j ) ( 2k + 2j) hence we derive the common binomial factorizable . This gives us: - (k + j )^2
06

Combine Factors

Combine the factors derived from full factored expression: ( k +j) + (k +2k) )\here final representation derived is: \ -\( k +j\)^2

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.

quadratic equations
A quadratic equation is a type of polynomial equation of the second degree. This means it has the highest exponent of 2. The general form of a quadratic equation is represented as: \[ ax^2 + bx + c = 0 \]where:
  • a is the coefficient of the squared term (x^2), and it cannot be zero.
  • b is the coefficient of the linear term (x).
  • c is the constant term.
Quadratic equations often appear in various real-world scenarios like projectile motion, area problems, and optimization problems.
factoring by grouping
Factoring by grouping is a method used to factor polynomials with four or more terms. The idea is to rearrange the polynomial and group terms in pairs, making it easier to factor out common terms. Here are the general steps:
  • Step 1: Group terms in pairs, usually by first arranging the polynomial in descending order of exponents.
  • Step 2: Factor out the common term from each pair.
  • Step 3: If done correctly, the leftover terms should share a common binomial factor, which can then be factored out.
Factoring by grouping is especially useful for polynomials that don't factor easily by other methods.
binomial factors
Binomial factors are expressions consisting of two terms, typically in the form \(a + b\) or \(a - b\). In our exercise, the factors we identify are binomials. Recognizing these patterns can make the factoring process more straightforward. For our exercise:
  • The expression was: \(-4k^2 - 5kj - j^2\)
  • We grouped and factored it as: \(-(k + j)^2\).
  • This indicates the common factors available in the polynomial.
Having binomial factors simplifies solving equations and understanding polynomial behavior.
algebraic expressions
Algebraic expressions are combinations of variables, constants, and operations. They are fundamental to algebra and come in various forms, including polynomials. Key points about algebraic expressions include:
  • Variables: Symbols that represent unknown values (e.g., \(k\) and \(j\) in our exercise).
  • Constants: Fixed numerical values within the expression (e.g., \(-4\) and \(-1\)).
  • Operations: Addition, subtraction, multiplication, and division.
Understanding how to manipulate and factor these expressions, including quadratic types, will make solving algebraic problems significantly easier.

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

Simplify each expression. $$ \frac{m-3}{m(2 m-6)} $$

Imagine rolling three regular dice and multiplying all three numbers. a. How many number triples are possible when you roll three dice? b. Without finding the products of every possible roll, describe a way you could determine whether an odd product or an even product is more likely. c. Use your method from Part b to determine whether an even product or an odd product is more likely.

Sports You have been asked to organize the matches for the singles competition at your local tennis club. There are seven players in the competition, and each must play every other player once. a. The matches Player A must play are listed below. Notice that this list already includes the match of Player A against Player B. Copy the table, and in the row for Player B, list all the other matches Player B must play. (In other words, list all the matches that include Player B but do not include Player A.) b. Predict the number of matches Player C must play that do not include Players A or B. Write your prediction in the 鈥淣umber of Matches鈥� column. c. Predict the number of matches for the remaining players. In each case, consider only those matches that do not include the players listed above that player. d. Check your predictions by listing the matches for each player. e. Describe the pattern in the 鈥淣umber of Matches鈥� column. Why do you think this pattern occurs? f. Find the total number of matches that will be played in the singles competition. g. Use what you have discovered in this problem to predict the total number of matches in a singles competition involving eight players. Explain how you found your answer.

Five seventh grade friends鈥擜nya, Ben, Calvin, Dan, and Ezra鈥� challenged five eighth grade friends鈥擵ic, Wendi, Xavier, Yvonne, and Zac鈥攖o a backgammon tournament. They put the names of the seventh graders into one hat and the names of the eighth graders into another. To determine the two players for each match, they pulled one name from each hat. a. What is the size of the sample space in this situation? That is, how many different pairs of names are possible? Explain. b. What is the probability that the next match will involve Anya and either Xavier or Yvonne? c. What is the probability that the next pair drawn will not involve Calvin? d. Suppose all 10 names are put into one hat, and two are drawn at random. What is the probability that the pair will include one seventh grader and one eighth grader? Explain.

In the Orange-and-White game, three white marbles and one orange marble are placed in a bag. A player randomly draws two marbles. If the marbles are different colors, the player wins a prize. a. List all the possible pairs in the sample space. (Hint: Label the marbles \(\mathrm{W} 1, \mathrm{W} 2\) \(\mathrm{W} 3,\) and \(\mathrm{O}\) .) b. What is the probability of winning a prize?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.