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Chapter 5: Problem 287

Why does \((a+b)^{2}\) result in a trinomial, but \((a-b)(a+b)\) result in a binomial?

Short Answer

Expert verified
(a+b)^{2} results in a trinomial because it expands to three terms. (a-b)(a+b) results in a binomial because it simplifies to two terms.

Step by step solution

01

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

To expand \( (a+b)^2 \), use the formula for the square of a binomial: \( (a+b)^2 = a^2 + 2ab + b^2 \). This results in three terms: \( a^2, 2ab, \) and \( b^2 \).
02

Confirm Trinomial Result for \( (a+b)^2 \)

Notice that the expansion of \( (a+b)^2 \) produces three distinct terms: a term with \( a^2 \), a term with \( 2ab \), and a term with \( b^2 \). Therefore, \( (a+b)^2 \) is a trinomial.
03

Expand \( (a-b)(a+b) \)

To expand \( (a-b)(a+b) \), use the difference of squares formula: \( (a-b)(a+b) = a^2 - b^2 \). This expression simplifies directly to two terms: \( a^2 \) and \( -b^2 \).
04

Confirm Binomial Result for \( (a-b)(a+b) \)

The expression \( a^2 - b^2 \) contains only two terms: \( a^2 \) and \( -b^2 \). Therefore, \( (a-b)(a+b) \) is a binomial.

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Key Concepts

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

trinomial
A trinomial is a polynomial with exactly three terms. It鈥檚 represented in the form: \( ax^2 + bx + c \). Each term can contain different powers of the same variable or even different variables.

For example, in the expression \( a^2 + 2ab + b^2 \), each part separated by a plus sign is a distinct term. The expansion of \( (a+b)^2 \) is a great example of forming a trinomial. Each of these three parts together makes up a trinomial.
binomial
A binomial is a polynomial consisting of exactly two terms connected by a plus or minus sign. Think of it as having two distinct parts.

For example, \( a + b \) is a binomial because of its two separate terms. Similarly, when you see an expression like \( a^2 - b^2 \), it is a binomial because it has exactly two terms.

In the exercise given, we see the binomial terms in the expression \( (a - b)(a + b) \) expanding into another binomial \( a^2 - b^2 \).
difference of squares
The difference of squares is a crucial algebraic identity. It states that \( (a-b)(a+b) \) simplifies to \( a^2 - b^2 \).

Here's why this works:
  • The product \( (a-b) \) and \( (a+b) \) leads to multiplying opposites, which gives us the middle terms that cancel each other out.
  • The term \( a \times a \) gives \( a^2 \) and the term \( b \times b \) gives \( b^2 \).
Hence, the remaining terms are just \( a^2 \) and \( -b^2 \), leading to the neat binomial form \( a^2 - b^2 \).
square of a binomial
When you square a binomial, you鈥檙e essentially multiplying it by itself. The formula for the square of a binomial \( (a + b)^2 \) is:
  • \( (a+b)(a+b) = a^2 + 2ab + b^2 \)
This expansion happens because:
  • First, you multiply \( a \) by \( a \) to get \( a^2 \).
  • Second, \( a \) by \( b \) and \( b \) by \( a \) each give \( ab \), which together give \( 2ab \).
  • Lastly, \( b \) multiplied by \( b \) results in \( b^2 \).
The presence of these three combined results forms a trinomial: \( a^2 + 2ab + b^2 \). Notice that the middle term is doubled because it involves both additions from the binomial multiplication.

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