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Chapter 5: Problem 46

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

Short Answer

Expert verified
The square of the binomial \((2 x+5 y)^{2}\) is \(4x^{2}+20xy+25y^{2}\)

Step by step solution

01

Write down the provided binomial

The provided binomial is \((2x+5y)^{2}\)
02

Apply the Binomial square formula

Using the formula \((a+b)^{2} = a^{2}+2ab+b^{2}\), where \(a=2x\) and \(b=5y\), this gives us \((2x)^{2}+2(2x)(5y)+(5y)^{2}\)
03

Simplify the expression

Simplify the above expression to obtain \(4x^{2}+20xy+25y^{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial expansion
The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. In simple terms, binomial expansion helps in breaking down expressions like \((a+b)^n\) into a series with simpler terms.

The formula for the square of a binomial, \((a+b)^2\), is \(a^2 + 2ab + b^2\). This means:
  • The first term is the square of the first part, \(a^2\).
  • The second term is twice the product of the two parts, \(2ab\).
  • The third term is the square of the second part, \(b^2\).
For example, if we have \((2x+5y)^2\), we identify \(a = 2x\) and \(b = 5y\), and apply the formula to simplify the expression.
Algebraic expressions
Algebraic expressions consist of variables, numbers, and operations. They form the backbone of many mathematical problems.

In our example, \((2x + 5y)^2\) is an algebraic expression. It can be broken down into basic parts:
  • "2x" and "5y" are terms. Each term consists of a coefficient and a variable.
  • When combined, these terms create a binomial.
Understanding how to manipulate and expand such expressions is crucial in solving a variety of algebraic problems.

By practicing how to expand binomials, you enhance your ability to tackle more complex algebraic challenges.
Polynomial multiplication
Multiplying polynomials is an essential skill in algebra that allows combining expressions to get simplified results.

In our example, the multiplication involves applying the binomial square formula to \((2x+5y)^2\). Here鈥檚 how the process unfolds:
  • First, compute the square of the first term: \((2x)^2 = 4x^2\).
  • Next, find twice the product of the two terms: \(2(2x)(5y) = 20xy\).
  • Finally, calculate the square of the second term: \((5y)^2 = 25y^2\).
Combining these results gives us the expanded polynomial: \(4x^2 + 20xy + 25y^2\).

Learning to perform polynomial multiplication helps in navigating through higher-level math topics, including calculus and advanced algebra.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{\left(2 a^{3}\right)^{5}}{\left(b^{4}\right)^{5}}$$

Will help you prepare for the material covered in the next section. Find the missing exponent, designated by the question mark, in the final step. $$ \frac{x^{7}}{x^{3}}=\frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}=x^{?} $$

Use a graphing utility to determine whether the divisions have been performed correctly. Graph each side of the given equation in the same viewing rectangle. The graphs should coincide. If they do not, correct the expression on the right side by using polynomial division. Then use your graphing utility to show that the division has been performed correctly. $$\frac{x^{3}+3 x^{2}+5 x+3}{x+1}=x^{2}-2 x+3$$

Exercises \(172-174\) will help you prepare for the material covered in the first section of the next chapter. In each exercise, find the product. $$4 x^{3}\left(4 x^{2}-3 x+1\right)$$

What is a trinomial? Give an example with your explanation.
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