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Chapter 4: Problem 79

Find each product. $$ [(5 x+1)+6 y]^{2} $$

Short Answer

Expert verified
25x^2 + 60xy + 10x + 36y^2 + 12y + 1

Step by step solution

01

Identify the Expression

The given expression is \( (5x + 1) + 6y \) raised to the power of 2. It can be represented as \[ \big((5x + 1) + 6y\big)^2 \].
02

Apply the Binomial Theorem

Recall the binomial theorem \[ (a + b)^2 = a^2 + 2ab + b^2 \]. In this case, \( a = (5x + 1) \, and \ b = 6y \).
03

Calculate \ a^2 \

Calculate \[ (a)^2 = (5x + 1)^2 \]. Using the binomial theorem again, \[ (5x + 1)^2 = (5x)^2 + 2(5x)(1) + (1)^2 = 25x^2 + 10x + 1 \].
04

Calculate \ 2ab \

Calculate \[ 2ab = 2(5x + 1)(6y) \]. Using the distributive property, \[ 2(5x + 1)(6y) = 2[30xy + 6y] = 60xy + 12y \].
05

Calculate \ b^2 \

Calculate \[ (b)^2 = (6y)^2 = 36y^2 \].
06

Combine All Terms

Add the results from Step 3, Step 4, and Step 5 together: \[ 25x^2 + 10x + 1 + 60xy + 12y + 36y^2 \]. Combining like terms, the final expanded form is \[ 25x^2 + 60xy + 10x + 36y^2 + 12y + 1 \].

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Key Concepts

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

Binomial Theorem
The binomial theorem is a powerful tool in algebra that helps to expand expressions raised to a power. For example, if you have \( (a + b)^2\), the binomial theorem tells us this can be expanded to \[a^2 + 2ab + b^2\]. This theorem works by breaking down the expression into simpler parts, which can then be easily calculated. It's very useful for expanding expressions without having to multiply everything out manually. In our example, we used the binomial theorem to first expand \( (5x + 1)^2\), and then we applied the theorem again to calculate other parts of our expression.
Distributive Property
The distributive property is a fundamental property of algebra that allows you to multiply a single term by each term inside a set of parentheses. For example, \[a(b + c) = ab + ac\]. In our problem, we used the distributive property to calculate \[2( (5x + 1)(6y) )\], which simplifies to \[2(30xy + 6y)\] and then \[60xy + 12y\]. This property makes it easier to handle algebraic expressions and is especially useful in combining terms or simplifying equations.
Combining Like Terms
Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. For example, \[3x\] and \[7x\] are like terms, but \[3x\] and \[3y\] are not. In our expression, we had several terms to combine: \[25x^2 + 60xy + 10x + 36y^2 + 12y + 1\]. After identifying and grouping the like terms, we were able to write the expanded form more concisely.

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