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Chapter 5: Problem 7

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

Short Answer

Expert verified
The product is \(x^2 + 4xy + 4y^2\).

Step by step solution

01

Write down the given expression

The given expression is \((x + 2y)^2\).
02

Use the square of a binomial formula

Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). In this problem, identify \(a = x\) and \(b = 2y\).
03

Substitute the values into the formula

Substitute \(a = x\) and \(b = 2y\) into the formula, getting: \[(x + 2y)^2 = x^2 + 2(x)(2y) + (2y)^2\].
04

Simplify the expression

First, calculate each term individually: \(x^2\) stays as it is, \(2(x)(2y) = 4xy\), and \((2y)^2 = 4y^2\).
05

Combine the simplified terms

Combine all the terms to get the simplified expression: \(x^2 + 4xy + 4y^2\).

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Key Concepts

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

Square of a Binomial
To understand binomial expansion, it's important to grasp the concept of squaring a binomial. A binomial is just an algebraic expression consisting of two terms; for example, \(a + b\). When we square a binomial, we multiply it by itself. So, \( (a + b)^2 = (a + b) \times (a + b) \).

However, there is a quicker way to handle this using a special formula. The formula is \( (a + b)^2 = a^2 + 2ab + b^2 \).

This means you don't need to multiply each term individually. Instead, you just square the first term, add twice the product of the two terms, and then square the second term.
Polynomial Expression
A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

For example, the expression \( x^2 + 4xy + 4y^2 \) is a polynomial because it is the sum of terms that follow this pattern: each term is the product of a coefficient (like \(4\)) and variables raised to a power (like \( x^2 \) or \( xy \)).

Polynomials can have constants, variables, and exponents, and they are used extensively in algebra. In our case, squaring the binomial \( (x+2y) \) results in the polynomial \( x^2 + 4xy + 4y^2 \).
Algebraic Simplification
Simplifying algebraic expressions means rewriting them in the simplest form. This involves combining like terms and making the expression as compact as possible.

Take the polynomial expression we derived: \( x^2 + 4xy + 4y^2 \). To simplify it, we should combine any like terms (though in this example, each term is unique).

  • First, write each term clearly.
  • Calculate each part: \(x^2\) is \(x^2\), \(2(x)(2y)\) simplifies to \(4xy\), and \( (2y)^2 = 4y^2 \).
  • Finally, combine them: \(x^2 + 4xy + 4y^2 \).


These small steps ensure your expressions are as clear and concise as possible, aiding in solving more complex problems down the line.

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Most popular questions from this chapter

Evaluate each expression for \(x=3\) $$ 4 x^{3}-5 x^{2}+2 x-5 $$

Use scientific notation to calculate the answer to each problem. In September of \(2009,\) the population of the United States was about 307.5 million. To the nearest dollar, calculate how much each person in the United States would have had to contribute in order to make one lucky person a trillionaire (that is, to give that person \(\$ 1,000,000,000,000) .\)

If it costs \(\$ 15\) to rent a chain saw, plus \(\$ 2\) per day, the binomial \(2 x+15\) gives the cost to rent the chain saw for \(x\) days. Evaluate this polynomial for \(x=6 .\) Use the result to fill in the blanks: If the saw is rented for ___\(-\) days, the cost is ___.

The polynomial equation $$ y=-0.0545 x^{2}+5.047 x+11.78 $$ gives a good approximation of the age of a dog in human years y, where \(x\) represents age in dog years. Each time we evaluate this polynomial for a value of \(x,\) we get one and only one output value y. For example, if a dogs is 4 in dog years, let \(x=4\) to find that \(y=31.1\) (lirify this, This means that the dogs is about 31 yr old in human years. This illustrates the concept of a function, one of the most important topics in mathematics. It used to be thought that each dog year was about 7 human years, so that \(y=7 x\) gave the number of human years for \(x\) dog years. Evaluate \(y\) for \(x=9,\) and interpret the result.

Perform each division. $$ \frac{-4 x+3 x^{3}+2}{x-1} $$
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