/*! 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 19 Factor each polynomial. SEE EXAM... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 19

Factor each polynomial. SEE EXAMPLE 1. (OB]ECTIVE 1) $$x^{2}-36$$

Short Answer

Expert verified
The factored form of the polynomial \(x^{2}-36\) is \((x+6)(x-6)\).

Step by step solution

01

Identify the squares

The polynomial can be recognized as a difference of perfect squares. From the given polynomial \(x^{2}-36\), it can be identified that \(x^{2}\) is the perfect square of \(x\) and 36 is the perfect square of 6.
02

Apply the formula

The formula for factoring the difference of squares is \(a^{2} - b^{2} = (a+b)(a-b)\). Using this formula, \(a\) would be \(\sqrt{x^{2}}= x\) and \(b\) would be \(\sqrt{36} = 6\).
03

Factor the difference

As per the formula of difference of squares, the factored form of polynomial will be \((x+6)(x-6)\). Hence, the factored form of the given polynomial \(x^{2}-36\) is \((x+6)(x-6)\).

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.

Difference of Squares
The difference of squares is a specific algebraic pattern used to simplify expressions that involve square terms. This pattern is recognized when you have an expression like \(a^2 - b^2\), which is characterized by two squares separated by a subtraction sign. In essence, the first term and the second term are both perfect squares.To factor a polynomial with this structure, you apply the difference of squares formula:
  • Formula: \( a^2 - b^2 = (a+b)(a-b) \)
  • The terms are rewritten as a product of a sum and a difference.
This technique is illustrated by the polynomial \(x^2 - 36\), where \(x^2\) is the square of \(x\) and \(36\) is the square of \(6\). Therefore, it factors to \((x+6)(x-6)\). This method is always applicable when you recognize this specific pattern.
Perfect Squares
Perfect squares are numbers or terms that are the result of an integer or a number being multiplied by itself. Identifying perfect squares is crucial in algebra, particularly when factoring polynomials.When given a number or variable expression, evaluate whether it is a perfect square:
  • \( x^2 \) is a perfect square because it results from multiplying \(x\) by itself.
  • \( 36 \) is a perfect square because \(6 \times 6 = 36\).
Recognizing perfect squares in an expression helps simplify complex algebraic operations, such as factoring using difference of squares or other polynomial factorizations.Understanding perfect squares allows you to quickly factor expressions like \(x^2 - 36\) by determining the square roots of each term and applying the difference of squares formula.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical relationships.Key aspects include:
  • Variables represent unknowns or quantities that can change, such as \(x\) in \(x^2 - 36\).
  • Constants are specific numbers with fixed values, like \(36\) in this expression.
  • Operators such as addition (+), subtraction (-), multiplication (脳), and division (梅) are used to manipulate these numbers and variables.
Algebraic expressions like \(x^2 - 36\) involve understanding how to manipulate symbols to solve or factor the expression. This understanding is foundational for breaking down complex expressions and solving algebra problems.

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. If the polynomial is prime, so indicate. $$3 b^{2}+3 a^{2}-a b$$

Factor. $$6 p^{2}-p q-2 q^{2}$$

Write the terms of each trinomial in descending powers of one variable. Then factor. $$15+8 a^{2}-26 a$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$-r^{2}+2 s^{2}+r s$$

Factor. $$a^{2}-4 a-32$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete 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.