/*! 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 130 Factor the polynomial. $$x^{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 130

Factor the polynomial. $$x^{2}+4 x-5$$

Short Answer

Expert verified
The factorized form of the given polynomial \( x^{2}+4x-5 \) is \( (x-1)(x+5) \)

Step by step solution

01

Identifying the Constants

In this polynomial, \( x^{2}+4x-5 \), the constants 'a', 'b' and 'c' are 1, 4 and -5 respectively.
02

Finding Factors

The next step involves identifying the two numbers, which when multiplied equal to -5 (product of 'a' and 'c') and when added equal to 4. The numbers satisfying these conditions are 5 and -1, because \(5 \times -1 = -5\), and \(5 + (-1) = 4\).
03

Writing the Polynomial in Factorized Form

The identified numbers are the coefficients for the factored form of the polynomial. So the factorized form of the polynomial \( x^{2}+4x-5 \) will be \( (x-1)(x+5) \)

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.

Understanding Quadratic Equations
Quadratic equations are a specific type of polynomial equation. They take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
This means that the equation includes a squared term (\(x^2\)) as its highest power.
Quadratic equations play a crucial role in algebra. They appear often in real-world problems. Solving these equations involves finding values of \(x\) that make the equation true. These values are known as the "roots" or "solutions" of the equation.
There are several methods to solve quadratic equations. Factoring, completing the square, and using the quadratic formula are popular options. Understanding the structure can help you choose the best method for solutions.
Exploring Factoring Techniques
Factoring is one of the key techniques used to solve quadratic equations. The goal is to write the quadratic equation as a product of two simpler expressions. This process makes finding the roots straightforward.
In the polynomial \(x^2 + 4x - 5\), the factors \((x-1)(x+5)\) represent the solution process. Here's how factoring usually works for quadratics:
  • Identify two numbers that multiply to \(a \times c\) and add to \(b\).
  • Rewrite the middle term using these numbers.
  • Factor by grouping the terms into pairs.
Factoring is effective when the quadratic can be easily expressed as the product of two expressions. However, it may not always be straightforward, especially with more complex equations.
The Role of Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and appear in many mathematical problems.
In a quadratic equation, expressions like \(x^2 + 4x - 5\) include:
  • Variables (\(x\))
  • Coefficients (numbers that multiply the variables, like 1 and 4)
  • Constants (numbers without variables, like -5)
Understanding algebraic expressions allows you to manipulate and simplify them. These skills are crucial for solving equations, performing calculations, and expressing complex ideas clearly.
Mastering algebraic expressions facilitates solving equations like quadratics and determining their factored forms, as seen in the exercise.

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 a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{aligned}&y_{1}=7\\\&y_{2}=2^{x-1}-5\end{aligned}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2$$

Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1.\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\log _{10} x=(x-3)^{2}$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0.5, 1.0), (2, 12.5), (4, 33.2), (6, 65.7), (8, 98.5),(10, 150.0)$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.