Problem 46 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(5 x-7)^{2}$$

Step-by-Step Solution

Answer

The expanded form is \(25x^2 - 70x + 49\).

Step 1: Identify the Binomial to Expand

The original expression is \((5x - 7)^2\), which needs to be expanded.

Step 2: Special Patterns Recognition: Square of a Binomial

Recognize this as a square of a binomial, \((a - b)^2\), which equals \(a^2 - 2ab + b^2\). Here, \(a = 5x\) and \(b = 7\).

Step 3: Apply the Formula

Using \((a - b)^2 = a^2 - 2ab + b^2\), replace \(a\) and \(b\) with \(5x\) and \(7\) respectively.

Step 4: Calculate Each Term

Calculate \((5x)^2 = 25x^2\), \(2(5x)(7) = 70x\), and \(7^2 = 49\).

Step 5: Combine the Terms for the Final Expression

Combine the above calculations to get the expanded form: \(25x^2 - 70x + 49\).

Key Concepts

Square of a Binomial

When multiplying binomials, especially when you have something like \((a - b)^2\), you're dealing with a square of a binomial.This is a special pattern in algebra that simplifies expansion using a straightforward rule.
You don't need to multiply it out step by step with distribution; instead, recognize the formula:

• \((a - b)^2 = a^2 - 2ab + b^2\)

This means you square the first term, subtract twice the product of both terms,then add the square of the second term.
For example, with \((5x - 7)^2\), you identify \(a\) as \(5x\) and \(b\) as \(7\).
It's essential to remember this pattern as it saves time and avoids errors.

Polynomial Expansion

Polynomial expansion is the process of simplifying a polynomial expression into a widely-readable form.For binomials, this often involves using the rules of algebraic identities or special patterns to expand expressions efficiently.
Rather than multiplying each term individually, shortcuts like the square of a binomial simplify the expansion process.For instance, evaluating \((5x - 7)^2\) using the square binomial formula \((a - b)^2 = a^2 - 2ab + b^2\),simplifies calculations immensely.
The expansion results in a polynomial, which can be written in the form

• \(25x^2 - 70x + 49\)

By substituting \(a\) and \(b\) in the identity with actual numbers and variables, you can easily perform polynomial expansion.

Special Patterns in Algebra

Special patterns in algebra, like the square of a binomial, help solve problems more succinctly than traditional multiplication.These patterns act as shortcuts, making the calculation process much more efficient.
In our example, \((5x - 7)^2\),taps into one of these patterns, showing how mathematical practice leverages well-established formulas.
Recognizing these patterns lets you avoid repetitive computations.

• The square of a sum: \((a + b)^2 = a^2 + 2ab + b^2\)
• The square of a difference: \((a - b)^2 = a^2 - 2ab + b^2\)
• The product of a sum and difference: \((a + b)(a - b) = a^2 - b^2\)

These identities simplify polynomial operations and are crucial for solving algebraic expressions efficiently.