Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((5-8 x)^{2}\)

Step-by-Step Solution

Answer

The product is \( 64x^{2} + 80x + 25 \).

Step 1: Identify the Components of the Expression

In the given expression, \( (5-8x)^{2} \), identify \( 5 \) as \( a \) and \( -8x \) as \( b \).

Step 2: Apply the Square Formula

Using the square formula \( (a-b)^{2} = a^{2} - 2ab + b^{2} \), substitute \( a = 5 \) and \( b = -8x \) into the formula to get \( (5)^{2} - 2*5*(-8x) + (-8x)^{2} \).

Step 3: Calculate the Resulting Expression

The resulting expression simplifies to \( 25 + 80x + 64x^{2} \).

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used to expand expressions that are raised to a power. It transforms them into a sum of terms without parentheses. When we have an expression like \((5 - 8x)^2\), polynomial expansion helps us rewrite it in a more understandable form.To expand a polynomial, you need to apply specific rules or formulas, such as the square or binomial theorems discussed below. The main goal is to simplify the expression by calculating each component step-by-step. This is crucial when dealing with more complex algebraic equations, as it allows easier substitution and evaluation.Polynomial expansion plays a vital role in algebra, and learning it lays the foundation for understanding more complex mathematical concepts. It turns a compact notation into something we can work with directly within calculations. Remember:

• Identify the terms and constants correctly - knowing these helps set up the expansion appropriately.
• Follow the expansion rules accurately - this ensures that each term is expanded as it should be.
• Simplify the resulting expression carefully - combine like terms and ensure accuracy in your final expression.

This method simplifies dealing with powers of algebraic expressions and makes problem-solving more manageable.

Square Formula

The square formula is a specific case of polynomial expansion that deals with squaring a binomial. A binomial is an algebraic expression with two terms. The square of a binomial involves using the formula:\[(a-b)^2 = a^2 - 2ab + b^2\]This formula helps us calculate the square of the entire expression by focusing on each part separately:

• \(a^2\): This is the square of the first term.
• \(-2ab\): This multiplies the two terms from the binomial, taking into account the sign between them.
• \(b^2\): This is the square of the second term.

In the exercise with \((5 - 8x)^2\), we set \(a = 5\) and \(b = -8x\). By substituting these values into the square formula, we calculate:

• \((5)^2 = 25\)
• \(-2 \times 5 \times (-8x) = 80x\)
• \((-8x)^2 = 64x^2\)

After computing each part, we combine them into the expanded form: \(25 + 80x + 64x^2\). This method ensures all components are accounted for and is essential for quickly and accurately expanding expressions with exponents of two.

Binomial Theorem

The Binomial Theorem is a generalized method for expanding expressions of the form \((a + b)^n\), where \(n\) is a positive integer. It provides a formula that helps expand powers of binomials easily. Although our example deals with squaring, which is a simpler case, understanding the binomial theorem is beneficial for broader applications.The theorem states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from \(n\), calculated as \(\frac{n!}{k!(n-k)!}\). Each term in the expansion has:

• \(a^{n-k}\): the power of the first term decreases as \(k\) increases.
• \(b^k\): the power of the second component increases as \(k\) increases.

While for \((5 - 8x)^2\), using the square formula is simpler, binomial theorem helps when \(n\) is greater. It allows efficient handling of expressions such as \((a + b)^3\), \((a + b)^4\), etc., giving each term within the expansion.The binomial theorem underscores the power of combinatorial mathematics in simplifying algebraic expansions, making them accessible and efficient.