Problem 3 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ b^{2}+10 b+25 $$

Step-by-Step Solution

Answer

The factorization of the expression is \((b+5)^2\).

Step 1: Identify pattern

Notice that this expression is a perfect square trinomial. A perfect square trinomial is a trinomial that results from squaring a binomial. In general, we can express a perfect square trinomial pattern in the form \((a+b)^2\), which will give us \(a^2+2ab+b^2\). In our given trinomial, we can rewrite it as \((b)^2+2*(b)*(5)+(5)^2\). Here, \(a\) corresponds to \(b\) and \(b\) corresponds to 5 in the square pattern.

Step 2: Factor the perfect square trinomial

The perfect square trinomial can be factored by the following rule: If the equation is in the form \(a^2+2ab+b^2\), it can be factored into \((a+b)^2\). So, factoring \(b^2+10b+25\) would result in \((b+5)^2\), because 5 when squared gives 25 and the multiplication of these values gives 10.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial. It comes from squaring a binomial, which means multiplying a binomial by itself.

When you see a trinomial of the form \(a^2 + 2ab + b^2\), you are looking at a perfect square trinomial.
This pattern occurs when a binomial like \((a+b)\) is squared. Mathematically, it looks like this:

• \((a+b)^2 = a^2 + 2ab + b^2\)

For example, in the given expression \(b^2 + 10b + 25\), you'll see that it fits this pattern:

• \(b^2\) is \(a^2\)
• \(10b\) is \(2ab\)
• \(25\) is \(b^2\)

Recognizing this pattern allows you to quickly identify that the expression comes from a squared binomial.

Binomial Square

A binomial square results from multiplying a two-term expression (binomial) by itself. Simply put, it is the product of \((a + b)(a + b)\).

This multiplication expands according to the distributive property. Here's how you expand a basic binomial squared:

• \((a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2\)

Breaking it down, you multiply each term in the first binomial by each term in the second binomial.

• \(a \times a = a^2\)
• \(a \times b = ab\)
• \(b \times a = ab\)
• \(b \times b = b^2\)

Adding these results gives \(a^2 + 2ab + b^2\). Recognizing such an outcome helps in identifying perfect square trinomials and simplifies the process of factoring polynomials.

Polynomial Factoring

Polynomial factoring is a technique used to express a polynomial as a product of simpler polynomials, akin to finding factors of a number. This process is crucial for simplifying expressions and solving equations.

Factoring a perfect square trinomial involves recognizing the square pattern and expressing it as the square of a binomial.

• For \(b^2 + 10b + 25\), identifying it as a perfect square trinomial lets us write it as \((b + 5)^2\).

Steps for factoring polynomials often include:

• Identify common factors or special patterns (like perfect squares).
• Break down the expression into these factors.

Once a polynomial is factored, solving or simplifying becomes much easier. Mastery of recognizing these patterns and techniques is essential in algebra and higher-level math courses.