/*! 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 64 (a) factor by grouping. Identify... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 64

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 2 p x-2 h x-p v+h v $$

Short Answer

Expert verified
The expression factors to \[ (p - h)(2x - v) \]. Both \[ (p - h) \] and \[ (2x - v) \] are prime polynomials.

Step by step solution

01

Rearrange the expression

Group the terms in pairs that share common factors. The expression is: \[ 2px - 2hx - pv + hv \]
02

Factor out the common factors

First, factor out common terms in each group. For \[2px - 2hx\], factor out \[2x\]: \[2x(p - h)\]. For \[-pv + hv\], factor out \[v\]: \[ -v(p - h) \]. Now the expression looks like: \[ 2x(p - h) - v(p - h) \]
03

Factor the common binomial

Factor \[ (p - h) \] out of the entire expression: \[ (p - h)(2x - v) \]
04

Identify any prime polynomials

Check if \[ (p - h) \] and \[ (2x - v) \] can be factored further. Since they are both already in their simplest form, they are prime polynomials.
05

Check by expanding

Expand \[ (p - h)(2x - v) \] back to verify: \[ 2x(p - h) - v(p - h) \]. This simplifies to: \[ 2px - 2hx - pv + hv, \] which matches the original expression.

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.

prime polynomials
When you hear the term 'prime polynomial,' think about prime numbers from basic arithmetic. Just like how prime numbers only have divisors of 1 and themselves, a prime polynomial is a polynomial that cannot be factored further using integer coefficients. In our example, the expressions \( (p - h) \) and \( (2x - v) \) are prime polynomials if they cannot be broken down into simpler factorable expressions. One quick way to identify a prime polynomial is to ensure it doesn't have any common factors or possible combinations that yield more basic polynomials. It's essentially in its simplest form already.
common factors
Understanding common factors is crucial when it comes to factoring by grouping. A common factor is a term that can divide two or more terms in a polynomial without leaving a remainder. For example, in the polynomial expression \(2px - 2hx - pv + hv\), you can see that \(2x\) is a common factor in the first pair \[2px - 2hx\], while \(-v\) is a common factor in the second pair \[pv + hv\]. Finding and isolating these common factors allows you to simplify the polynomial by reclining them as individual terms. This practice lays the groundwork for easier factorization.
rearrange expression
Rearranging the expression is a critical first step. It helps you to see the common factors more clearly. Once you have the original polynomial, group the terms that have common factors together. For instance, starting with the expression \(2px - 2hx - pv + hv\), we can group it as \[2px - 2hx\] and \[-pv + hv\]. By doing this, we merely make it simpler to identify and factor out the common elements in subsequent steps. Always read through the polynomial carefully and systematically group the terms before trying to factor them out.
factor common binomial
The last step in factoring by grouping involves factoring out the common binomial. After rearranging the expression and identifying common factors, you often end up with something like \(2x(p - h) - v(p - h)\). Notice how \(p - h\) is repeated? This repetition allows us to factor it out of the entire expression altogether. So, we take \(p - h\) out, resulting in the factorized form \((p - h)(2x - v)\). This step not only simplifies the polynomial but checks if it can be factored further. If the inside terms are prime, which in this case they are, then your work is done for factoring by grouping.

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

Factor completely. Identify any prime polynomials. $$ 6 a^{2} b+2 a^{2} c-24 b-8 c $$

(a) solve. (b) check. $$ c^{2}-8 c+16=0 $$

Factor completely. Identify any prime polynomials. $$ 9 x^{3}+36 y^{3} $$

The formula for density, \(d\), is \(d=\frac{m}{v}\), where \(m\) is the mass and \(v\) is the volume. The density of a steel sphere is \(7.85 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\), and its mass is \(5 \times 10^{2} \mathrm{~g}\). Solve the formula for \(v\), and find the volume of this sphere. Round to the nearest whole number.

(a) solve. (b) check. $$ z^{2}=11 z-18 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.