/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 287 Why does \((a+b)^{2}\) result in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Multiply the binomials using (a) the Distributive Property; (b) the FOIL method; (c) the Vertical Method. $$ (7 q+4)(3 q-8) $$

Simplify each expression using the properties for exponents. (a) \(d^{3} \cdot d^{6}\) (b) \(4^{5 x} \cdot 4^{9 x} (c) 2 y \cdot 4 y^{3}\) (d) \(w \cdot w^{2} \cdot w^{3}\)

Use synthetic Division to find the quotient and remainder. $$ 2 x^{3}-11 x^{2}+16 x-12 \text { is divided by } x-4 $$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Simplify each expression. \(\left(\frac{k^{-2} k^{8}}{k^{3}}\right)^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.