Problem 26 - Elementary and Intermediate Algebra

Question

Factor. $$ d^{2}-10 d+25 $$

Step-by-Step Solution

Answer

The expression factors to \( (d - 5)^2 \).

Step 1: Recognize the form of a perfect square trinomial

The given expression is \( d^2 - 10d + 25 \). It matches the form of a perfect square trinomial, \( a^2 - 2ab + b^2 \), where \( a = d \) and \( b = 5 \), as \( (5)^2 = 25 \) and \( 2 \times d \times 5 = 10d \).

Step 2: Write as a binomial squared

Since the expression is a perfect square trinomial, we can write it as the square of a binomial. Thus, \( d^2 - 10d + 25 = (d - 5)^2 \).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. The general form is:

• \( a^2 + 2ab + b^2 \) or
• \( a^2 - 2ab + b^2 \)

Both formats result in a simplified binomial squared. In such trinomials, the first term and the last term are perfect squares, and the middle term is twice the product of the square root of the first and last terms.

For instance, in the expression \( d^2 - 10d + 25 \), the first term \( d^2 \) is the square of \( d \), and the last term 25 is the square of 5. The middle term, -10d, is equal to \( -2 \times d \times 5 \). Because of these characteristics, this particular trinomial can be rewritten as a binomial square: \((d - 5)^2\). Identifying perfect square trinomials allows for efficient factoring, transforming more complex expressions into simpler, manageable forms.

Binomial

A binomial is an algebraic expression containing two distinct terms. It's a subset of polynomials and can be as simple as \( x + 3 \) or more complex as \( 3a^2 - 5b \). When working with quadratic expressions, recognizing binomials becomes crucial, especially when dealing with perfect square trinomials.

To convert a perfect square trinomial into a binomial squared, one must recognize the two terms involved. Taking our example, \( d^2 - 10d + 25 \) can be simplified into the binomial \( (d - 5) \). Thus, the expression can be written as \( (d - 5)(d - 5) \), which simplifies further to \((d - 5)^2\). Recognizing and manipulating binomials is foundational in algebra, easing the process of solving, simplifying, and factoring complex expressions.

Algebraic Expressions

Algebraic expressions include variables, constants, and arithmetic operations. They form the building blocks of algebra and are used to represent real-world scenarios mathematically.

In a typical algebraic expression like \( d^2 - 10d + 25 \), we have:

• Variables such as \( d \)
• Constants like 25
• Operations including addition and subtraction

These expressions can range from simple monomials like \( 7x \) to more complex polynomials. Factoring these expressions involves transforming them into products of simpler expressions. For example, factoring the expression \( d^2 - 10d + 25 \) involves identifying that it's a perfect square trinomial and then rewriting it as \((d - 5)^2\). Understanding the structure and components of algebraic expressions is key to performing operations such as factoring, expanding, and solving equations in algebra.