Problem 54 - College Algebra

Question

Factor each perfect square trinomial. $$ 25 x^{2}+10 x+1 $$

Step-by-Step Solution

Answer

Therefore, the factored form of the perfect square trinomial \(25x^{2} + 10x + 1\) is \((5x + 1)^{2}\).

Step 1: Recognize the Pattern

First of all, remind the formula: \(a^{2} + 2ab + b^{2} = (a + b)^{2}\), the given expression \(25x^{2} + 10x + 1\) has the same pattern which represents \(a = 5x\), \(b = 1\). So, it could be simplified to \((5x + 1)^{2}\). It is based on the fact that the square of a binomial \((a + b)^{2}\) equals \(a^{2} + 2ab + b^{2}\).

Step 2: Check Your Work

Check the solution by expanding \((5x + 1)^{2}\) as \((5x + 1) \cdot (5x + 1)\). Use the distributive property to multiply each term in the first binomial with each term in the second binomial, and then add the products, you should get the original trinomial \(25x^{2} + 10x + 1\). If the check is successful, then the factorization is correct.

Key Concepts

Factoring Trinomials

Factoring trinomials involves writing a quadratic expression as the product of two binomials. This process can be straightforward if the trinomial is a perfect square trinomial, because it follows a specific pattern. For instance, the trinomial \(25x^2 + 10x + 1\) is a perfect square trinomial.

Recognizing the perfect square trinomial pattern is crucial: \(a^2 + 2ab + b^2 = (a + b)^2\). In our example:

• \(a = 5x\)
• \(b = 1\)

By identifying these terms, we can factor \(25x^2 + 10x + 1\) into \((5x + 1)^2\), which simplifies the task significantly. This approach is particularly useful because it reduces a quadratic expression into a product of two identical binomials, making it easier to solve or further manipulate in algebraic operations.

By checking the factorization through expansion, you ensure that all steps were executed correctly.

Binomial Expansion

Binomial expansion is the process of multiplying a binomial by itself one or more times. It uses the distributive property to compute the expanded expression. In the context of perfect square trinomials, expanding \((5x + 1)^2\) involves multiplying \((5x + 1)\) by itself.

To expand:

• Multiply \(5x\) by \(5x\) to get \(25x^2\).
• Then, multiply \(5x\) by \(1\) (twice) to get \(5x\), and another \(5x\) again which results in \(10x\) as these are added together.
• Finally, multiply \(1\) by \(1\) to obtain \(1\).

Combine these products to return to the original expression: \(25x^2 + 10x + 1\). Effectively mastering binomial expansion not only helps in rechecking factorization work but also forms a fundamental building block in higher algebraic concepts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis of algebra and gracefully mix constants and variables through addition, subtraction, multiplication, and more.

In the case of our perfect square trinomial, \(25x^2 + 10x + 1\) represents an algebraic expression where:

• \(25x^2\) is the quadratic term, reflecting how the variable \(x\) is squared.
• \(10x\) is the linear term, indicating a direct relation multiplied by a constant.
• \(1\) is the constant term, which remains unchanged by the variable.

Understanding algebraic expressions is key to solving equations, factoring, and many other algebraic tasks. As you delve deeper into algebra, becoming comfortable with expressions allows you to more easily manipulate and solve mathematical problems, driving towards a higher mastery in mathematics.