/*! 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 16 Find each product. $$ x(2 x+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 16

Find each product. $$ x(2 x+5)^{2} $$

Short Answer

Expert verified
The product is \(4x^3 + 20x^2 + 25x\).

Step by step solution

01

- Recognize the Given Expression

The given expression to find the product is \( x(2x + 5)^2 \). This is a polynomial multiplication problem.
02

- Apply the Binomial Theorem

Use the binomial theorem to expand \((2x + 5)^2\). According to the binomial theorem, \((a + b)^2 = a^2 + 2ab + b^2\). Here, let \(a = 2x\) and \(b = 5\).
03

- Expand the Binomial

Expand \((2x + 5)^2\): \[(2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2\] Calculate these individually: \[(2x)^2 = 4x^2\] \[2(2x)(5) = 20x\] \[5^2 = 25\] Now combine them: \[4x^2 + 20x + 25\]
04

- Distribute the Monomial

Now multiply the expanded binomial \(4x^2 + 20x + 25\) by the monomial \(x\). This means multiplying each term in the polynomial by \(x\): \[x \times (4x^2 + 20x + 25)\] This gives: \[4x^3 + 20x^2 + 25x\]

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 theorem
The binomial theorem is a fundamental concept in algebra that provides a way to expand expressions that are raised to a power. For example, \( (a + b)^n \) can be expanded using the theorem, which states that:
\[ (a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{n}b^n \]
This can look complex, but it's straightforward with practice.
For instance, if we have \( (2x + 5)^2 \), we can expand it as follows:
  • Identify 'a' and 'b': Here, 'a' is 2x and 'b' is 5.
  • Apply the General Formula: According to the binomial theorem,
    \[ (2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2 \]
  • Calculate:
    \[ (2x)^2 = 4x^2 \]
    \[ 2(2x)(5) = 20x \]
    \[ (5)^2 = 25 \]
Combining these results, we get
\[ 4x^2 + 20x + 25 \]
With understanding the binomial theorem, expanding polynomials becomes much easier!
polynomial expansion
Polynomial expansion is a process used in algebra to simplify expressions where terms consist of variables raised to a power. When working with polynomial expressions like \( (2x + 5)^2 \), the goal is to 'expand' this into a simple expression.
Here is how we do this:
  • Start with the Binomial: Look at the polynomial we need to expand, in this case \( (2x + 5)^2 \).
  • Apply the Binomial Theorem: Expand using the theorem: \[ (2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2 \]
  • Calculate Individual Terms: Calculate each term separately:
    \[ (2x)^2 = 4x^2 \]
    \[ 2(2x)(5) = 20x \]
    \[ (5)^2 = 25 \]
  • Combine Terms: Add up the calculated terms to get the expanded polynomial:
    \[ 4x^2 + 20x + 25 \]
This expansion makes it easier to understand and solve polynomial problems. The simplified form gives a clear view of all individual terms involved.
distributive property
The distributive property is essential when multiplying polynomials. It states that multiplying a number by a sum is the same as multiplying the number by each term in the sum and then adding the products.
Mathematically, this is expressed as:
\[ a(b + c) = ab + ac \]
This principle helps break down complex multiplications like \( x(4x^2 + 20x + 25) \). Here's how you apply it step-by-step:
  • Identify the Monomial and Polynomial: In our example, the monomial is 'x' and the polynomial is \( (4x^2 + 20x + 25) \).
  • Distribute the Monomial: Multiply 'x' by each term in the polynomial:
    \[ x \times 4x^2 = 4x^3 \]
    \[ x \times 20x = 20x^2 \]
    \[ x \times 25 = 25x \]
  • Combine Results: Put together the products to get:
    \[ 4x^3 + 20x^2 + 25x \]
Using the distributive property simplifies and organizes polynomial multiplication, making it manageable and clear. The property is not limited to numbers; it applies broadly, making it a versatile tool in algebra.

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

Use scientific notation to calculate the answer to each problem. See Examples 3-5. During the \(2007-2008\) season, Broadway shows grossed a total of \(9.38 \times 10^{8}\) dollars. Total attendance for the season was \(1.23 \times 10^{7} .\) What was the average ticket price for a Broadway show? (Source: The Broadway League.)

Perform each indicated operation. Find the difference between the sum of \(5 x^{2}+2 x-3\) and \(x^{2}-8 x+2\) and the sum of \(7 x^{2}-3 x+6\) and \(-x^{2}+4 x-6\)

Pollux, one of the brightest stars in the night sky, is 33.7 light-years from Earth. If one light-year is about \(6,000,000,000,000\) mi, about how many miles is Pollux from Earth? (Source: World Almanac and Book of Facts.)

Each statement contains a number in boldface italics. Write the number in scientific notation. At the end of 2007 , the total number of cellular telephone subscriptions in the world reached about \(3,305,000,000 .\) (Source: International Telecommunications Union.)

Each statement contains a number in boldface italics. Write the number in scientific notation. In \(2007,\) the leading U.S. advertiser was the Procter and Gamble Company, which spent approximately 5,230,000,000 dollars. (Source: Crain Communications, Inc.)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.