Problem 11 - Elementary and Intermediate Algebra

Question

Find each product. See Example 1. $$ (m-6)^{2} $$

Step-by-Step Solution

Answer

The product is \(m^2 - 12m + 36\).

Step 1: Understand the Problem

The problem asks us to find the product of the expression \((m - 6)^2\). This means we need to expand this binomial squared and determine its product.

Step 2: Apply the Square of a Binomial Formula

The square of a binomial formula is \((a - b)^2 = a^2 - 2ab + b^2\). In our problem, let \(a = m\) and \(b = 6\). Apply the formula: \((m - 6)^2 = m^2 - 2(m)(6) + 6^2\).

Step 3: Simplify Each Term

Calculate each term from the formula applied: - The first term is \(m^2\).- The second term is \(-2(m)(6) = -12m\).- The third term is \(6^2 = 36\).

Step 4: Write the Final Expression

Combine all the simplified terms from Step 3 to get the final expression of the product: \(m^2 - 12m + 36\).

Key Concepts

Expanding Binomials

When you encounter a binomial expression, such as \((x + y)\) or \((m - 6)\), and you need to multiply it by itself or another binomial, the process is called expanding binomials. Expanding is a method where we utilize distributive properties to eliminate parentheses and rewrite the expression as a polynomial.

In a simple example: \((a + b)(a + b)\), we apply distributive property and multiply each term in the first binomial by each term in the second binomial:

• Multiply the first term of the first binomial by both terms in the second binomial.
• Multiply the second term of the first binomial by both terms in the second binomial.

This results in: \(a*a + a*b + b*a + b*b = a^2 + 2ab + b^2\).
Expanding is fundamental for simplifying and rewriting expressions that allow for easier interpretation and manipulation in algebraic equations.

Square of a Binomial

The square of a binomial involves taking a binomial, such as \((a + b)\) or \((m - 6)\), and raising it to the power of 2, namely \((a + b)^2\) or \((m - 6)^2\). This process leverages a special algebraic identity that saves time compared to using distributive properties from scratch each time.

The identity used is:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• Similarly, \((a - b)^2 = a^2 - 2ab + b^2\)

This means you just need to square the first term, double the product of the two terms, and square the last term. In the context of our problem with \((m - 6)^2\), using \(a = m\) and \(b = 6\) results in:

• \(m^2 - 2(m)(6) + 6^2 = m^2 - 12m + 36\)

Recognizing and applying this formula greatly simplifies the process of expanding squared binomials and is a highly efficient algebraic tool.

Polynomial Expressions

Polynomial expressions are algebraic expressions that involve sums and differences of terms consisting of variables raised to positive integer powers and their coefficients. Understanding and working with polynomial expressions is a pillar of algebra.

For instance, after expanding a binomial, like in the problem \((m - 6)^2\), you get a simplified polynomial expression: \(m^2 - 12m + 36\). Each part of a polynomial, such as \(m^2\), \(-12m\), or \(36\), is a term.

Key points about polynomial expressions include:

• The degree of a polynomial is determined by the highest power of the variable. In \(m^2 - 12m + 36\), it is 2.
• Polynomials can be classified by their number of terms: monomials (one term), binomials (two terms), and trinomials (three terms).

By understanding the structure and properties of polynomial expressions, you can manipulate and solve them in a wide range of algebraic problems, laying a solid foundation for further studies in mathematics.