/*! 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 52 Factor completely. \(v^{2}-11 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 52

Factor completely. \(v^{2}-11 v x+24 x^{2}\)

Short Answer

Expert verified
The completely factored form of the expression is \((v - 3x)(v - 8x)\).

Step by step solution

01

- Identify the Quadratic Coefficient, Linear Coefficient, and Constant Term

Recognize that the given expression is a quadratic trinomial of the form \[ av^2 + bvx + cx^2 \] Here, the coefficients are as follows: \( a=1 \), \( b=-11 \), \( c=24 \).
02

- Find Factors of the Constant Term (24) that Add up to the Linear Coefficient (-11)

Look for two numbers that multiply to 24 (the product of \( a \times c \)) and add up to -11 (the linear coefficient). The numbers are -3 and -8, since \(-3 \times -8 = 24\) and \(-3 + -8 = -11\).
03

- Rewrite the Middle Term Using the Factors Found

Rewrite the middle term \( -11vx \) using the factors we found, -3 and -8: \[ v^2 - 3vx - 8vx + 24x^2 \].
04

- Factor by Grouping

Group the terms in pairs and factor out the greatest common factor from each pair: \[ v(v - 3x) - 8x(v - 3x) \].
05

- Factor out the Common Binomial

Notice the common binomial factor \(v - 3x\) in each term and factor that out: \[ (v - 3x)(v - 8x) \].

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.

quadratic equation
A quadratic equation is a second-degree polynomial equation in a single variable, typically in the form: \[ ax^2 + bx + c = 0 \]where:
  • a is the quadratic coefficient (not equal to zero),
  • b is the linear coefficient,
  • c is the constant term.
In our example, we started with the quadratic trinomial, \[ v^2 - 11vx + 24x^2\].Here, the quadratic coefficient \(a\) is 1, the linear coefficient \(b\) is -11, and the constant term \(c\) is 24.
factoring by grouping
Factoring by grouping is a method used to factor polynomials that have four terms. The key is to group the terms in pairs and then factor out the common factors. Here鈥檚 how we applied it:
  • We started with the expression \[ v^2 - 3vx - 8vx + 24x^2 \].
  • We grouped them as: \[ (v^2 - 3vx) + (-8vx + 24x^2) \].
  • We factored out the greatest common factor (GCF) from each group: \[ v(v - 3x) - 8x(v - 3x) \].
Finally, we observed that \(v - 3x\) is a common term and factored it out, resulting in: \[ (v - 3x)(v - 8x)\]. Factoring by grouping simplifies complex expressions by breaking them down into manageable chunks.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, or division). In this problem, we dealt with the algebraic expression \[ v^2 - 11vx + 24x^2 \]. To solve it, we performed operations on the expression:
  • We first identified the coefficients and constant terms.
  • Then we found suitable factors for splitting the middle term.
  • Finally, grouping helped us to simplify the expression.
Understanding how to manipulate algebraic expressions is essential for solving a wide range of math problems.
coefficients
Coefficients are the numerical or constant multipliers in terms of an algebraic expression. In a quadratic equation, coefficients play a vital role in determining the nature of the roots and the shape of the graph of the equation. For example, in the quadratic trinomial
  • In \[ v^2 - 11vx + 24x^2 \], the coefficient of \(v^2\) is 1.
  • The coefficient of \(vx\) is -11.
  • The constant term (or the coefficient of \(x^2\)), is 24.
Coefficients help us in factoring and solving quadratic equations through various methods, like factoring by grouping, completing the square, or using the quadratic formula. Recognizing and working with coefficients is a fundamental skill in algebra.

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

Which is the correct factored form of the given polynomial? \(3 a^{2}-5 a-2\) A. \((3 a+1)(a-2)\) B. \((3 a-1)(a+2)\)

Find product. \((5 z+2)(3 z-2)\)

An expression is factored when it is written as a product, not a sum. Which of the following are not factored? $$ \left(2 k^{2}+5 k\right)+1 $$

Identify each monomial as a perfect square, a perfect cube, both of these, or neither of these. (a) \(64 x^{6} y^{12}\) (b) \(125 t^{6}\) (c) \(49 x^{12}\) (d) \(81 r^{10}\)

Factor each trinomial completely. $$ x^{2}-8 x+16 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.