/*! 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 46 In Exercises 15鈥�58, find each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 46

In Exercises 15鈥�58, find each product. $$ (x-4)^{2} $$

Short Answer

Expert verified
\(x^2 - 8x + 16\)

Step by step solution

01

Identify the Binomial Terms

In this problem, the binomial is \((x-4)\) and it is being squared. In the algebraic principle, this corresponds to \(a = x\) and \(b = 4\).
02

Apply the Algebraic Principle

Now substitute these values into the principle \((a-b)^2 = a^2 - 2ab + b^2\). This gives us \((x - 4)^2 = x^2 - 2*x*4 + 4^2\).
03

Simplify Further

This simplifies to \(x^2 - 8x + 16\).
04

Write Down the Final Answer

The final result of \((x-4)^2\) is \(x^2 - 8x + 16\) after completing the simplification.

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.

Binomial Expansion
Binomial expansion is a fundamental concept in algebra where we take a binomial, a sum or difference of two terms, and expand it into a polynomial. When expanding a binomial, it often involves squaring or raising it to a higher power. The formula for expanding a squared binomial is given by:
  • \((a+b)^2 = a^2 + 2ab + b^2\)
  • \((a-b)^2 = a^2 - 2ab + b^2\)
These formulas come in handy whenever you need to square a binomial expression like \((x-4)^2\). Recognizing the structure and using these templates helps solve problems quickly and accurately. By substituting the identified terms into the appropriate format, we can easily perform the calculations and reach the expanded polynomial.
Squared Binomials
A squared binomial refers to an expression where a binomial term is raised to the power of two. In essence, you鈥檙e multiplying the binomial by itself. Let鈥檚 consider the binomial \((x-4)^2\). To square it, you use the format \((a-b)^2 = a^2 - 2ab + b^2\), where in this case, *a* represents *x* and *b* is *4*.

Squaring of this binomial consists of three key steps:
  • Calculate \(a^2\): Here, \(x^2\) is the result when *x* is squared.
  • Calculate \(-2ab\): Multiply *x* by *4* and by *-2*, resulting in \(-8x\).
  • Calculate \(b^2\): Multiply 4 by itself to get \(16\).
Finally, by summing the results from the individual components, we derive the resulting expression \(x^2 - 8x + 16\). This complete item represents the squared binomial.
Polynomial Simplification
Polynomial simplification involves reducing a polynomial to its simplest form. This process requires combining like terms and performing arithmetic operations to streamline the expression. In the problem of squaring \((x-4)\), after applying the binomial expansion, we arrive at an un-simplified polynomial: \(x^2 - 2 \cdot x \cdot 4 + 4^2\).

Here's how simplification works:
  • Calculate each term separately: \(x^2\), \(-8x\), and \(+16\).
  • Check if there are any like terms. Here, each term is distinct: \(x^2\), \(x\), and constant term are all unique.
  • Combine all the terms into a final result: \(x^2 - 8x + 16\). This is the simplest form for the expanded binomial.
Simplifying expressions helps not only in obtaining the correct answer but also plays a crucial role in solving greater algebraic equations effectively.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some rational numbers are not positive.

Factor completely. $$-x^{2}-4 x+5$$

Explain how to add \(\sqrt{3}+\sqrt{12}\)

Factor Completely. $$y^{7}+y$$

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.