/*! 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 42 Factor completely. If the polyno... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 42

Factor completely. If the polynomial cannot be factored, write prime. \(m^{2}+10 m-30\)

Short Answer

Expert verified
The polynomial is prime.

Step by step solution

01

Identify and write down the polynomial

The given polynomial is -30.
02

Look for factor pairs

Determine pairs of integers that multiply to the constant term (-30) and add up to the coefficient of the middle term (10). The coefficient of the middle term is 10, and the constant term is -30.
03

Verify factor pairs

Check the sums of all factor pairs: 1. (-1)(30), sum: 29 2. (1)(-30), sum: -29 3. (-2)(15), sum: 13 4. (2)(-15), sum: -13 5. (-3)(10), sum: 7 6. (3)(-10), sum: -7 7. (-5)(6), sum: 1 8. (5)(-6), sum: -1 No factor pairs sum up to 10.
04

Conclude primality

As no factor pairs of -30 can sum up to 10, the polynomial cannot be factored further.

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 an expression into a product of simpler expressions called 'factors'. Factoring is often used to simplify polynomials.
For example, consider the polynomial \(x^2 + 5x + 6\). To factor this, we look for two numbers that multiply to the constant term (6) and add up to the coefficient of the middle term (5).
These numbers are 2 and 3. So, the factors of \(x^2 + 5x + 6\) are \((x + 2)(x + 3)\).
Factoring helps solve equations, simplify expressions, and is essential in calculus and number theory.
  • Identify the terms in the polynomial.
  • Find pairs of numbers that multiply to the constant term.
  • Check if their sum matches the coefficient of the middle term.
  • If they do, rewrite the polynomial as a product of factors.
polynomials
Polynomials are algebraic expressions made up of terms combined using addition, subtraction, and multiplication. Each term consists of a constant coefficient multiplied by variables raised to whole-number exponents.
For example, \(2x^3 - 4x^2 + 3x - 1\) is a polynomial with four terms. Polynomials are often classified by their degree, which is the highest power of the variable in the expression.
  • A polynomial of degree 2 is called a quadratic polynomial (e.g., \(x^2 + 3x + 2\)).
  • A polynomial of degree 3 is called a cubic polynomial (e.g., \(x^3 - 2x^2 + x - 5\)).
  • Degrees help identify the polynomial's behavior and other properties.
Understanding polynomials and their properties is crucial for algebra and higher-level math courses.
prime polynomial
A prime polynomial is a polynomial that cannot be factored further over the integers. Just as a prime number has no divisors other than 1 and itself, a prime polynomial only has trivial factors.
For example, consider the polynomial \(x^2 + x + 1\). There are no two integers that multiply to 1 and add up to 1, so this polynomial is prime.
To determine if a polynomial is prime:
  • Attempt to factor it.
  • If you cannot find valid factor pairs that satisfy the requirements, then the polynomial is prime.
In our exercise, we concluded that \(m^2 + 10m - 30\) is a prime polynomial because no integer pairs multiply to -30 and add up to 10.
integer pairs
Integer pairs are pairs of whole numbers used in the factoring process to find two numbers that multiply to a specific product and add up to a specific sum. They are crucial for factoring polynomials.
For example, to factor \(x^2 + 7x + 12\), we look for pairs of integers that multiply to 12 and add up to 7. These pairs are: (3, 4) and (-3, -4).
  • Check all potential pairs of integers that multiply to the constant term.
  • Test each pair by adding them to see if they match the coefficient of the middle term.
  • Pairs that satisfy both conditions are used to write the polynomial in factored form.
This method allows you to systematically approach the factoring process and confirms whether a polynomial is factorable or prime.

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

Find the value of the indicated variable. Find \(b\) so that \(x^{2}+b x+25\) factors as \((x+5)^{2}\)

Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 9 x^{2}(r+3)^{3}+12 x y(r+3)^{3}+4 y^{2}(r+3)^{3} $$

Factor polynomial. \((3 m-n) k^{2}-13(3 m-n) k+40(3 m-n)\)

Concept Check The number 9 is a double solution of the equation \((x-9)^{2}=0 .\) Why is this so?

Fill in the blank with the correct response. The equation \(x^{3}+x^{2}+x=0\) is not a quadratic equation, because ______.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.