/*! 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 85 Explain how to factor \(x^{2}+8 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 85

Explain how to factor \(x^{2}+8 x+15\)

Short Answer

Expert verified
The factored form of the trinomial \(x^{2}+8 x+15\) is \((x+3)(x+5)\)

Step by step solution

01

Identify Coefficients

In the given expression \(x^{2}+8 x+15\), the coefficients are 1 (in front of \(x^2\)), 8 (in front of x) and 15 (the constant term).
02

Find Two Numbers

Look for two numbers that multiply to 15 (the third coefficient/the constant) and sum to 8 (the second coefficient). The numbers that fit this criteria are 3 and 5. Because 3*5=15 and 3+5=8.
03

Factor the Trinomial

The trinomial is factored as \((x+3)(x+5)\), because both when multiplied together give the original trinomial \(x^{2}+8 x+15\), and when added together yield 8x.

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.

Trinomial
A trinomial is a specific type of polynomial that consists of three distinct terms. In mathematical expressions, each term may include variables and coefficients. The standard format for a trinomial generally is expressed as:
  • A variable squared term, such as \(x^2\)
  • A linear variable term, such as \(8x\)
  • A constant, such as 15
These components come together to form the expression \(x^2 + 8x + 15\). Understanding the structure of trinomials is essential before diving into the factoring process.Recognizing that there are three terms is crucial, as each term plays a role in various algebraic operations, including factoring.
Coefficients
In the expression \(x^2 + 8x + 15\), coefficients are the numerical factors that multiply the variables in each term. They provide important information about the term's overall value:
  • The coefficient of the squared term \(x^2\) is 1. Often implicit, but equally essential.
  • The linear term \(8x\) has a coefficient of 8.
  • The constant term \(15\) serves as both the third term and a coefficient, though it lacks variables.
Understanding coefficients is a vital skill in algebra. They help us decide how terms relate and which values we're searching for during the factoring process.In solving and manipulating quadratic expressions, coefficients guide calculations, showing us how different parts of the expression contribute to its algebraic identity.
Factoring Process
The goal of factoring a trinomial expression like \(x^2 + 8x + 15\) is to rewrite it as a product of two binomials. This process breaks down into manageable steps to identify key values:
  • First, analyze the constant term, 15, looking for two numbers that multiply to it. Think of 3 and 5 since \(3 \times 5 = 15\).
  • The next step is to verify that these numbers add up to the middle coefficient, 8. Verify by calculating \(3 + 5 = 8\).
  • Once identified, the expression can be rewritten as \((x + 3)(x + 5)\), because these numbers satisfy the conditions.
This process illuminates the simplicity in breaking down quadratic expressions and how these smaller factors reconstruct the original expression.
Quadratic Expressions
Quadratic expressions form the backbone of many algebraic concepts and are characterized by the highest variable being squared, such as in \(x^2 + 8x + 15\). They usually come in the form of \(ax^2 + bx + c\):
  • "\(a\)" represents the coefficient of the squared term \(x^2\).
  • "\(b\)" is the coefficient of the linear term \(x\).
  • "\(c\)" is the constant or the coefficient without variables.
The importance of understanding these structures is twofold: first, they give us the foundation to recognize which algebraic methods to use, and second, they provide insight into possible solutions. By knowing how to manipulate these expressions鈥攍ike factoring and expanding鈥攚e gain powerful tools for navigating through more complex algebraic problems, making sense of their components and relationships in a larger mathematical framework.

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

The alligator, at one time an endangered species, is the subject of a protection program. The formula $$P=-10 x^{2}+475 x+3500$$ models the alligator population, \(P,\) after \(x\) years of the protection program, where \(0 \leq x \leq 12 .\) Use the formula to solve. After how long is the population up to \(5990 ?\)

In Exercises \(130-133,\) use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{3}-1=(x-1)\left(x^{2}-x+1\right)$$

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(x(x-4)=21\)

Use the \(x\)-intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=(x-2)(x+3)-6\) to solve \((x-2)(x+3)-6=0\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.