Question

\((1.7-3.2 t)^{2}\)

Short answer

The expanded form is \( 2.89 - 10.88t + 10.24t^2 \)

Step-by-step solution

Identify the expression

The expression given is \( (1.7 - 3.2t)^2 \). This is a binomial squared.

Apply the Binomial Theorem

The binomial theorem states that \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( a = 1.7 \) and \( b = 3.2t \).

Compute \(a^2\)

Calculate \( a^2 \): \((1.7)^2 = 2.89 \).

Compute \(2ab\)

Calculate \( 2ab \): \((2 \times 1.7 \times 3.2t) = 10.88t \).

Compute \(b^2\)

Calculate \( b^2 \): \( (3.2t)^2 = 10.24t^2 \).

Combine the results

Combine all parts: \( 2.89 - 10.88t + 10.24t^2 \).

Key concepts

binomial squaring

In algebra, squaring a binomial involves raising a two-term expression to the power of two. The binomial squaring follows a special formula from the binomial theorem. This is crucial because it simplifies the process and avoids mistakes in multiplication.

Let's take a closer look at our example: \( (1.7 - 3.2t)^2 \). Here, we have two terms inside the parentheses, namely 1.7 and \-3.2t. When we square this, the binomial theorem tells us to use the formula: \((a - b)^2 = a^2 - 2ab + b^2\).

This formula is very straightforward to use:

• Firstly, square the first term (\(a\))
• Secondly, double the product of both terms
• Lastly, square the second term (\(b\))

By following these steps carefully, we break down complex algebraic expressions into smaller, manageable parts.

polynomial expansion

Polynomial expansion is the process of expressing a polynomial as a sum of its terms. This is particularly important in algebra when dealing with expressions like binomials.

Given the binomial \( (1.7 - 3.2t)^2 \), our task is to expand it fully. Using the binomial theorem, we can compute each part step by step:

• First, calculate \( a^2 = (1.7)^2 = 2.89 \)
• Next, find \( 2ab = 2 \times 1.7 \times 3.2t = 10.88t \)
• Finally, evaluate \( b^2 = (3.2t)^2 = 10.24t^2 \)

Combining these, we get the fully expanded polynomial \( 2.89 - 10.88t + 10.24t^2 \). This expanded form is useful for various applications, including graphing functions or integrating polynomials.

algebraic expressions

An algebraic expression is a mathematical phrase that can involve numbers, variables, and operators like addition or multiplication.

In the context of our exercise, \( (1.7 - 3.2t)^2 \) is an algebraic expression, specifically a binomial. When we expand this expression, we apply known algebraic rules to simplify it. Understanding each term's role in the expression helps make complex problems more approachable.

Here, the expanded form \( 2.89 - 10.88t + 10.24t^2 \) gives us a clearer view of each component. Each term in this expanded form has a specific meaning and function:

• The constant term \( 2.89 \) is a fixed number.
• The linear term \( -10.88t \) includes the variable \( t \) to the power of one.
• The quadratic term \( 10.24t^2 \) includes the variable \( t \) squared.

By breaking down these terms, we can better understand and solve algebraic problems more efficiently.