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Chapter 5: Problem 41

Find each product. Does \((a+b)^{2}\) equal \(a^{2}+b^{2}\) in general? Explain.

Short Answer

Expert verified
\((a + b)^2\) does not equal \(a^2 + b^2\) in general because of the missing \(2ab\) term.

Step by step solution

01

Expand \((a+b)^{2}\)

Use the distributive property to expand \( (a + b)^{2} \). This can be written as \( (a + b)(a + b) \).
02

Apply the FOIL Method

Apply the FOIL (First, Outer, Inner, Last) method to distribute each term:\[ (a + b)(a + b) = a^{2} + ab + ba + b^{2} \]
03

Combine Like Terms

Since \( ab = ba \), combine the middle terms:\[ a^{2} + ab + ab + b^{2} = a^{2} + 2ab + b^{2} \]
04

Compare \( (a+b)^{2} \) and \( a^{2} + b^{2} \)

Compare the expanded form \( a^{2} + 2ab + b^{2} \) with the expression \( a^{2} + b^{2} \). The term \( 2ab \) is missing in \( a^{2} + b^{2} \).
05

Conclude

Therefore, \( (a+b)^{2} \) does not equal \( a^{2} + b^{2} \) in general. The additional term \( 2ab \) is essential.

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra. It states that for any numbers or expressions, the following holds true: \(a(b + c) = ab + ac\).
In the given problem, we apply the distributive property to \((a + b)^{2}\).

This can be rewritten as \((a + b)(a + b)\), which means we distribute every term in the first binomial to every term in the second.
Breaking it down: \[ (a + b)(a + b) \rightarrow a(a + b) + b(a + b) \], when distributed.
This gives us \(a^{2} + ab + ba + b^{2}\).
It simplifies and sets the stage for the next step in expanding the algebraic expression.
FOIL method
The FOIL method is a specific technique to multiply two binomials. It stands for First, Outer, Inner, Last. Each word denotes the terms to multiply:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the product.
  • Inner: Multiply the inner terms in the product.
  • Last: Multiply the last terms in each binomial.

Using the expression given: \((a + b)(a + b)\), we apply FOIL:
  • First: \(a \times a = a^{2}\)
  • Outer: \(a \times b = ab\)
  • Inner: \(b \times a = ba\)
  • Last: \(b \times b = b^{2}\)

Combining all these products: \[a^{2} + ab + ba + b^{2}\], we see that the terms \(ab\) and \(ba\) are similar, illustrating the need to combine like terms in the next step.
combining like terms
Combining like terms is essential to simplify expressions. Like terms are terms with the same variables raised to the same power.
From the expansion \(a^{2} + ab + ba + b^{2}\), we see that \(ab\) and \(ba\) are considered like terms because they both contain one \(a\) and one \(b\).
Using the commutative property of multiplication \(ab = ba\), we can sum these terms.
Therefore, \(ab + ba = 2ab\).
So, the original expanded form simplifies to \[a^{2} + 2ab + b^{2}\].
This shows how combining like terms gives us a clearer, simplified expression.
In summary, \((a + b)^{2}\) expands to \[a^{2} + 2ab + b^{2}\],
which is not equivalent to \[a^{2} + b^{2}\] due to the missing \(2ab\) term.

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