Problem 30 - Algebra A Combined Function

Question

Multiply. $$ (5 b-4)^{2} $$

Step-by-Step Solution

Answer

\((5b - 4)^2 = 25b^2 - 40b + 16\).

Step 1: Understanding the Problem

We need to multiply the binomial \((5b - 4)\) by itself, i.e., we are calculating \((5b - 4)^2\). This means \((5b - 4)\) multiplied by \((5b - 4)\).

Step 2: Applying the Binomial Theorem

Recall that \((a - b)^2 = a^2 - 2ab + b^2\). Here, our \(a\) is \(5b\) and \(b\) is \(4\), so we'll apply this to expand the expression \((5b - 4)^2\).

Step 3: Calculate Each Term

First, calculate \((5b)^2 = 25b^2\). Second, calculate \(-2 \cdot 5b \cdot 4 = -40b\). Third, calculate \(4^2 = 16\).

Step 4: Combining the Terms

Combine all the calculated terms from Step 3 using the binomial expansion: \((5b)^2 - 2\cdot5b\cdot4 + 4^2 = 25b^2 - 40b + 16\).

Step 5: Final Expression

The expression \((5b - 4)^2\) expands to \(25b^2 - 40b + 16\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. An example of an algebraic expression is \(5b - 4\). Here, '5' and '-4' are coefficients, with '5' multiplying the variable \(b\). Variables represent unknown values and can change.

When we manipulate algebraic expressions, we perform operations like addition, subtraction, multiplication, and division.

• Manipulation allows us to simplify or solve expressions and equations that include variables.
• These expressions often form the basis for constructing equations, which can then be solved to find the value of the variable(s).

The key to dealing with algebraic expressions is to carefully apply the rules of arithmetic and algebra while respecting the properties of operations sensitive to the expressions.

Binomial Theorem

The Binomial Theorem is a fundamental theorem in algebra that provides a formula for expanding powers of binomials. A binomial is a simple algebraic expression that has two terms, like \(a - b\). The theorem gives us a way to expand \((a - b)^n\) or \((a + b)^n\) without multiplying the binomial by itself multiple times.

For instance, using the Binomial Theorem,

• You can expand \((5b - 4)^2\) by identifying \(a = 5b\) and \(b = 4\).
• The expansion for \((a - b)^2\) is \(a^2 - 2ab + b^2\), simplifying the process of multiplying \(5b - 4\) by itself.

This theorem is particularly useful for dealing with higher powers of binomials due to its systematic approach.

Exponents

Exponents are an essential mathematical concept used to describe how many times a number, or base, is multiplied by itself. In the expression \((5b - 4)^2\), the '2' is the exponent that indicates the base \((5b - 4)\) should be multiplied by itself once.

Here are a few important points about exponents:

• They simplify expressions by using powers instead of repeated multiplication.
• Exponent rules, such as the power of a product or power of a power, help in simplifying expressions.

In our current exercise, calculating \((5b)^2\) requires knowledge of how to apply exponents properly, simplified as \(25b^2\).

Polynomials

Polynomials are algebraic expressions composed of more than one term, with each consisting of a number known as the coefficient and a variable raised to an exponent. An example of a polynomial is \(25b^2 - 40b + 16\), which results from expanding the original binomial expression \((5b - 4)^2\).

Attributes of polynomials include:

• They can have constants, variables, and exponents that are non-negative integers.
• The degree of a polynomial is the highest exponent present in the expression.
• Different types of polynomials include monomials, binomials, and trinomials based on the number of terms.

In this exercise, we find a trinomial polynomial that expresses the expanded form of the square of a binomial.