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Chapter 4: 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} because of the missing term 2ab.

Step by step solution

01

Identify the Expression

Start with the given expression \( (a+b)^{2} \) and question whether it equals \( a^{2}+b^{2} \).
02

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

Use the formula for the square of a binomial: \[ (a+b)^{2} = a^{2} + 2ab + b^{2}. \]
03

Compare Both Expressions

Compare the expanded form \(a^{2} + 2ab + b^{2}\) with the expression \(a^{2} + b^{2} \. Notice that the term \2ab \) is missing in \(a^{2} + b^{2} \).
04

Conclusion

Since \(a^{2} + 2ab + b^{2} \) is not equal to \(a^{2} + b^{2} \, the original statement is false in general. \ (a+b)^{2} \) does not equal \(a^{2} + b^{2} \).

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Key Concepts

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

quadratic expressions
Quadratic expressions involve variables raised to the power of two. They appear in the form of ax^2 + bx + ca, where a, b, and c are constants. The expression under consideration (a+b)^2 fits this form, though without a linear term. Expanding (a+b)^2 using algebraic rules gives us a^2 + 2ab + b^2. This forms a perfect square trinomial.
algebraic identities
Algebraic identities are fundamental truths in algebra, holding true for all values of the variables involved. One key identity is the square of a binomial: (a+b)^2 = a^2 + 2ab + b^2. This identity shows why (a+b)^2 eq a^2 + b^2. The missing term 2ab in a^2 + b^2 is crucial for the equality to hold. Understanding these identities helps students avoid common mistakes and simplifies complex algebraic problems.
polynomial expansion
Polynomial expansion involves expressing a polynomial in its extended form. For example, expanding the binomial (a+b)^2 gives the polynomial a^2 + 2ab + b^2. This process uses the distributive property and commutative properties of addition and multiplication. By expanding polynomials, we make implicit relationships within the polynomial explicit, enabling easier manipulation and understanding.
misconceptions in algebra
Misconceptions in algebra, like thinking (a+b)^2 = a^2 + b^2, trip up many students. This mistake overlooks the middle term 2ab, which results from the distributive property during expansion. Students often mistakenly simplify expressions too much or misuse algebraic identities. To avoid these errors, it's crucial to follow expansion steps carefully and understand the underlying algebraic principles. Practice and verification of results help solidify correct understanding.

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Most popular questions from this chapter

To understand how the special product \((a+b)^{2}=a^{2}+2 a b+b^{2}\) can be applied to a purely numerical problem. The number 35 can be written as \(30+5 .\) Therefore, \(35^{2}=(30+5)^{2} .\) Use the special product for squaring a binomial with \(a=30\) and \(b=5\) to write an expression for \((30+5)^{2} .\) Do not simplify at this time.

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{3,400,000,000(0.000075)}{0.00025} $$

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.)

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ \left(2 x+\frac{2}{3} y\right)\left(3 x-\frac{3}{4} y\right) $$

Evaluate. $$ 1000(1.53) $$
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