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Chapter 10: Problem 22

Write the square of the binomial as a trinomial. $$(a+8)^{2}$$

Short Answer

Expert verified
The square of the binomial \((a+8)^2\) as a trinomial is \(a^2 + 16a + 64\).

Step by step solution

01

Identify the Terms

In the binomial \(a+8\), a is the first term and 8 is the second term.
02

Plug into the formula

Now, we will square the binomial using the formula \((a+b)^2 = a^2 + 2ab + b^2\). So, plug a and 8 into the formula.
03

Calculate Expression

Calculating, we get \(a^2 + 2*a*8 + 8^2\). This Simplifies to \(a^2 + 16a + 64\).

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Key Concepts

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

Binomial
A binomial is an algebraic expression containing two terms that are added or subtracted. A classic example is \(a + b\). Here \(a\) and \(b\) are terms.
  • Terms: Parts of the expression separated by addition or subtraction.
  • Example: In \(3x + 4\), "3x" and "4" are the terms.
Binomials are important in algebra as they form the basis for more complex expressions and operations. Understanding how to manipulate these is crucial for solving equations and performing expansions.
Trinomial
A trinomial is an algebraic expression made up of three terms. These terms can involve variables raised to various powers or constants. Trinomials often result from operations involving binomials.
  • Terms: An expression with three parts, such as \(a^2 + b^2 + c\).
  • Example: \(x^2 + 3x + 2\).
Learning to identify and work with trinomials is essential for tasks like factoring or expanding expressions, which are significant skills in algebra.
Square of a Binomial
The square of a binomial is when you multiply a binomial by itself. This is important because it leads to a quadratic expression, specifically a trinomial.
  • Formula: \((a+b)^2 = a^2 + 2ab + b^2\).
  • Steps to expand: Identify terms and apply the formula.
  • Example: \((a + 8)^2\) expands to \(a^2 + 16a + 64\).
This concept is frequently used to simplify expressions and solve equations in algebra, providing a foundation for more advanced mathematics.
Polynomial Expansion
Polynomial expansion involves expressing a multiplied polynomial in its simplest form. This often requires using binomial expansions, such as the binomial theorem, to convert expressions into a standard polynomial form.
  • Processes: Utilize formulas like \((a+b)^n\) to expand.
  • Applications: Solve equations, distribute coefficients, and find roots.
The ability to expand polynomials is a key part of algebra; it helps in graphing functions, finding their roots, and solving complex equations efficiently.

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