Problem 30 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the components of the binomial

In the expression \((a+8)^2\), the components of the binomial are \(a\) and \(8\)

Step 2: Apply the binomial square formula

Use the formula \((x+y)^2 = x^2 + 2xy + y^2\), where \(x=a\) and \(y=8\). This leads to a squared term, a cross term, and a squared term.

Step 3: Calculate the values

The squared term of \(x=a\) is \(a^2\). The cross term of \(2xy = 2*a*8 = 16a\). The squared term of \(y=8\) is \(8^2 = 64\)

Step 4: Combine the terms

The original binomial squared \((a+8)^2\) is equivalent to the trinomial \(a^2 + 16a + 64\)

Key Concepts

Binomial Expansion

A binomial expansion involves the process of expanding an expression that is raised to a power, where the expression consists of two terms, often separated by a plus or minus sign. These expressions look like this:

• For example, \((a + b)^n\) where \(a\) and \(b\) are numbers or variables, and \(n\) is a positive integer.

The binomial theorem provides a systematic way to expand these expressions.
The expanded form is

• \((a + b)^2 = a^2 + 2ab + b^2\) when \(n=2\),
• which means we multiply the binomial by itself.

This expansion uses each term of the binomial repeatedly while following a pattern dictated by the powers and coefficients based on Pascal’s Triangle.
By understanding this, you can tackle binomials of higher powers too without directly multiplying the terms many times.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are important components in solving mathematical equations and understanding binomials.
Examples of algebraic expressions are:

• \(a + 8\), where \(a\) is a variable and \(8\) is a constant.
• Expressions like \(3x + 4y - 5\), involving more than one variable.

When working with algebraic expressions, you follow rules of arithmetic like addition, subtraction, multiplication, and division along with algebra-specific rules, like collecting like terms.
This allows you to simplify and solve various mathematical problems effectively, including expanding binomials.

Polynomial Expansion

Polynomial expansion takes an expression with multiple terms and breaks it down into simpler terms. This is useful for multiplication and allows easier manipulation of expressions.
A polynomial expression such as

• \((a+8)^2\)

can be decomposed through expansion by applying known algebraic identities or formulas.
In our exercise's solution:

• The square of the binomial \((a+8)^2\) was expanded to become a polynomial \(a^2 + 16a + 64\).

This process helps in identifying components like squared terms and cross terms, making it simpler to understand and compute additions or multiplications in polynomial form.
Breaking down polynomials in such a systematic manner allows you to handle large expressions efficiently.