/*! 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 59 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 59

Perform the indicated operations. $$(3 p+5)^{2}$$

Short Answer

Expert verified
\((3p + 5)^2 = 9p^2 + 30p + 25\).

Step by step solution

01

Understand the Expression

The exercise requires you to expand the expression \((3p + 5)^2\). This is a binomial squared expression which can be expanded using the binomial formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = 3p\) and \(b = 5\).
02

Apply the Binomial Formula

Substitute \(a = 3p\) and \(b = 5\) into the binomial formula. The formula becomes: \((3p)^2 + 2(3p)(5) + 5^2\).
03

Calculate Each Term

- Compute \((3p)^2 = 9p^2\).- Compute \(2(3p)(5) = 30p\).- Compute \(5^2 = 25\).
04

Combine All Terms

Combine the calculated terms: \(9p^2 + 30p + 25\). This is the expanded form of the given 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.

Understanding the Binomial Theorem
The binomial theorem provides a method for expanding expressions like \((a+b)^n\). When we encounter an expression such as \((3p + 5)^2\), the binomial theorem helps us expand it efficiently without having to multiply the binomial by itself manually. The specific case of \((a+b)^2\) follows the formula:
  • \(a^2\)
  • \(2ab\)
  • \(b^2\)
This means that instead of writing \((a+b) \, (a+b)\) and finding the product term by term, we can use this formula to directly find the expanded form. By substituting \(a = 3p\) and \(b = 5\) into this formula, we quickly calculate the expanded form as \( (3p)^2 + 2 \cdot (3p) \cdot 5 + 5^2 = 9p^2 + 30p + 25\). This method saves time and reduces the risk of error.
Expanding Expressions
Expression expansion is a common problem in algebra that involves breaking down expressions into a simpler or more usable form. With binomials, this usually means taking terms within parentheses and expanding them using the binomial theorem or simple distribution. When expanding a binomial squared, like \((3p+5)^2\), we apply the binomial formula, making it easier to handle complex expressions:
  • First, apply the individual terms: \((3p)^2\), which yields \(9p^2\).
  • Next, compute the middle term: \(2 \cdot 3p \cdot 5\), which calculates to \(30p\).
  • Finally, determine the square of the last term, \(5^2\), which gives us \(25\).
The final expanded expression is simply the sum of these results: \(9p^2 + 30p + 25\). Through practice, this process becomes intuitive and is fundamental to mastering algebra.
Polynomial Operations Made Simple
Polynomial operations are foundational in algebra, encompassing a series of techniques used to simplify, add, subtract, and multiply polynomial expressions. In this exercise, we have a polynomial that is generated by expanding a binomial expression.
  • The first step is to identify each individual term in the expression.
  • Maneuvering polynomials often involves combining like terms, which are terms sharing the same variables with identical exponents.
In the expression \(9p^2 + 30p + 25\),
  • there is a quadratic term \(9p^2\), a linear term \(30p\), and a constant \(25\).
The concept of polynomial operations extends beyond expansion to manipulation and simplification of complex polynomials through operations such as addition, subtraction, and further multiplication, often preparing the expressions for equation solving or graphing. Understanding these operations unlocks the ability to tackle more advanced algebra concepts smoothly.

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

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\sqrt[3]{2}}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{45}$$

Write each expression in radical form. Assume that all variables represent positive real numbers. $$(-m)^{2 / 3}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$-\sqrt[4]{32}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{\frac{5}{3 p}}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.