/*! 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 11 Factor completely. $$ 144 \t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 11

Factor completely. $$ 144 \times 2-1 $$

Short Answer

Expert verified
The expression simplifies to 287, which is a prime number and cannot be factored further.

Step by step solution

01

Simplify the Expression

Start by simplifying the expression. Multiply 144 by 2 to get 288. Therefore, the expression becomes \( 288 - 1 \).
02

Calculate the Difference

Subtract 1 from 288 to simplify the expression further. This results in \( 287 \).
03

Verify if the Result Can Be Factored

Check if the number 287 can be factored further by testing if it is a prime number. Verify potential divisibility by smaller prime numbers like 2, 3, 5, 7, and 11. None of these divide 287 perfectly. Thus, 287 is a prime number.

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.

Factoring
Factoring is the process of breaking down a mathematical expression into simpler components that can be multiplied to produce the original expression. In this exercise, we started with an expression: \(144 \times 2 - 1\). The goal is to determine if the result can be factored further. To factor successfully, follow these steps:
  • Look for a common factor in the terms of your expression, if there are multiple terms.
  • If the result is a single number, see if it can be split into a product of other numbers.
In this case, after simplifying the expression, we obtained 287. Therefore, our task is to determine if 287 can be broken down into smaller numbers that multiply to give 287, which leads us to the next steps involving simplification and prime testing.
Simplification
Simplification involves performing basic arithmetic operations to reduce an expression to its most straightforward form. Our initial expression was \(144 \times 2 - 1\).Here's how we simplified it:
  • Multiply \(144\) by \(2\) to get \(288\).
  • Subtract \(1\) from \(288\) to arrive at \(287\).
After simplification, our original expression was reduced to the number 287. This process not only makes it easier to work with numbers, but it also prepares the way for further factorization or simplification steps, such as checking for primality.
Prime Number Verification
Prime number verification is the process of determining whether a given number is a prime. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To verify if 287 is a prime number:
  • Check if it is divisible by small prime numbers such as 2, 3, 5, 7, and 11.
  • If none of these numbers divide 287 perfectly (i.e., no remainder), 287 is a prime number.
For 287, testing divisibility by these primes showed that none of them divide 287 without a remainder. Thus, we confirm that 287 is indeed a prime number, meaning it cannot be factored further. Understanding this process is crucial for recognizing prime numbers and simplifying expressions efficiently.

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 height of a projectile launched upward at a speed of 16 feet/second from a height of 192 feet is given by the function \(h(t)=-16 t 2+16 t+192\). How long will it take to hit the ground?

The diagonal of a rectangle measures 10 inches. If the width is 2 inches less than the length, then find the area of the rectangle.

The height of a triangle is 1 unit more than the length of its base. If the area is 5 units more than four times the height, then find the length of the base and height of the triangle.

The product of two consecutive positive integers is 42 . Find the integers.

Factor completely. $$ 5 m 2 n 2+10 m n-15 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Decision 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.