Problem 55 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$25 y^{2}-10 y+1$$

Step-by-Step Solution

Answer

The perfect square trinomial \(25y^2 - 10y + 1\) can be factored as \((5y - 1)^2\).

Step 1: Identify the coefficient of each term

The coefficients for the trinomial \(25y^2 - 10y + 1\) are 25, -10, and 1 respectively.

Step 2: Check if it's a perfect square trinomial

A perfect square trinomial conforms to the formula \(a^2 - 2ab + b^2\). To find a and b, we have to find the square roots of the first and the last terms which are: \(\sqrt{25y^2}\) = \(5y\) and \(\sqrt{1}\) = 1. We then substitute \(5y\) for a and 1 for b in the equation. Using the formula, we get \(5y^2 - 2*5y*1 + 1^2\). Simplifying gives \(25y^2 - 10y + 1\), which matches exactly with the original trinomial. Therefore, the equation is indeed a perfect square trinomial.

Step 3: Factor the trinomial

Using the connection between the formula \(a^2 - 2ab + b^2\) and binomial square \((a-b)^2\), we can write the trinomial as the square of a binomial. So, the trinomial \(25y^2 - 10y + 1\) can be factored as \((5y - 1)^2\).

Key Concepts

Polynomial Factoring

Understanding the process of polynomial factoring is crucial for solving algebraic expressions and equations. Factoring a polynomial involves breaking down the complex expression into simpler, multiplied factors, much like finding the original pieces of a puzzle. This technique is not just about simplifying an equation; it is also a strategy that prepares us for solving equations and understanding the properties of graphs of polynomial functions.

For instance, the expression provided in the exercise, 25y^2 - 10y + 1, is a trinomial that we aim to factor. The steps to factorize may vary depending on the type of polynomial, but generally, one looks for a common factor, patterns like the difference of squares, perfect square trinomials, or the use of the quadratic formula when applicable. Recognizing patterns is crucial and can greatly simplify the process. For this particular trinomial, it follows the pattern of a perfect square trinomial, which leads us to our next essential topic.

Perfect Square Trinomial Formula

A perfect square trinomial is a special form of polynomial that can be conveniently factored if one knows the pattern. The general perfect square trinomial formula is \(a^2 \pm 2ab + b^2\) and it factors into \(a \pm b)^2\). This pattern is easily identifiable because the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

In the given exercise \(25y^2 - 10y + 1\), by identifying the square roots of the first and last term as \(5y\) and \(1\), respectively, we can then recreate the middle term as twice the product of \(5y\) and \(1\), which gives us \(10y\), confirming it fits the perfect square trinomial structure. To ensure we're on the right track, always check that the middle term in your trinomial matches the formula's middle term; that is, it should be exactly twice the product of the square roots of the first and last term.

Binomial Square

A binomial square emerges from squaring a binomial expression. This action naturally creates a perfect square trinomial, as seen in the formula \(a \pm b)^2 = a^2 \pm 2ab + b^2\). When a binomial is squared, it results in the first term being squared, two times the product of both terms, and the second term squared.

The crux of solving the given exercise lies in recognizing that the trinomial in question is the result of a binomial square. By doing so, we avoid the longer process of guessing and checking or other factoring methods, and directly write the original trinomial \(25y^2 - 10y + 1\) as \(5y - 1)^2\), where \(5y\) is the square root of \(25y^2\) and \(1\) is the square root of \(1\), thus neatly factoring the expression. Having a solid understanding of binomial squares can greatly simplify polynomial factoring and is a powerful tool in algebra.