Problem 38 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$3 y^{3}-75 y$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(3 y^{3}-75 y\) is \(3y (y + 5)(y - 5)\).

Step 1: Identify common factor

First, identify the common factors in the given polynomial. In \(3 y^{3}-75 y\), the common factor is '3y'.

Step 2: Write down the common factor

Write down the common factor as part of a product. In this case, '3y' is written first, followed by an open bracket: \(3y (\)

Step 3: Write down the remaining factors

Each term of the polynomial is divided by the common factor, '3y'. This results in \(3y (y^{2} - 25)\)

Step 4: Factorize the difference of squares

Next we will factorize the difference of squares. The difference of two squares, \(a^{2} - b^{2}\) can be factored into \((a + b)(a - b)\). Here \(a = y\), and \(b = 5\). This leads to \(3y (y + 5)(y - 5)\). Hence the polynomial is fully factorized.

Key Concepts

Common Factors in Polynomials

Understanding common factors in polynomials is crucial for simplifying and solving algebraic equations. Much like finding common ground between friends to make plans, finding common factors in a polynomial means looking for terms that appear in every part of the expression.

Let's put on our math goggles and investigate the polynomial from our example, \(3 y^{3}-75 y\). If we envision the terms as two separate containers, our job is to see what mathematical 'items' both containers hold. These items can be numbers, variables, or a combination of both. In this case, both terms have '3' and 'y' as repeating items. So, we can confidently say that \(3y\) is the common factor.

By extracting this common factor and placing it in front of a set of parentheses, we set the stage to reveal what's left. The remaining factors are the content of the polynomial that has been stripped off the common item. Thus, by taking out \(3y\), we're simplifying the polynomial to its core, making it easier to manage and understand. This simplification is the first step in factoring the polynomial completely.

Difference of Squares

The phrase 'difference of squares' might sound like an architectural critique, but in the context of algebra, it's a pattern where two perfect squares are subtracted from one another, creating the form \(a^{2} - b^{2}\). To recognize this pattern, imagine you're looking at a tiled floor where each tile is a square. If you have one large square area and subtract a smaller square area from it, what you're left with fits the pattern of the difference of squares.

In our polynomial example after factoring out the common factor \(3y\), we stumbled upon the expression \(y^{2} - 25\), which is a classic example of the difference of squares because it resembles the pattern \(a^{2} - b^{2}\), where our 'tiles' are \(y^{2}\) and \(25\) (since \(25\) is the square of \(5\)).

But why is this recognition helpful? Because it lets us crack open this expression further. By applying the formula for factoring a difference of squares, \((a + b)(a - b)\), we transform \(y^{2} - 25\) into \((y + 5)(y - 5)\). This opens up the expression, revealing a more refined factorization, akin to a neatly segmented chocolate bar, easy to break apart and digest!

Polynomial Factorization

The process of breaking down polynomials into products of simpler factors is like deconstructing a recipe to its core ingredients. Polynomial factorization can turn a daunting, complex algebraic dish into a sequence of more straightforward steps that are easier to digest and solve.

When embarking on the journey of factorization, like a detective piecing together clues, we systematically search for patterns, common factors, and identities that simplify the expression. For \(3y^{3} - 75y\), we brought out the common factor, then spotted and used the difference of squares to further simplify into \(3y(y + 5)(y - 5)\).

This polynomial has been elegantly served up as a product of its factors, similar to presenting a beautifully plated meal. By breaking the expression down in this manner, we make it much easier for someone working with the equation to understand its structure and solve it. As in cooking, where different ingredients combine to create a dish, in algebra, these factors can sometimes recombine to solve for a variable or to simplify an equation further. The art of polynomial factorization is a powerful tool in the algebraic kitchen and serves as a foundation for solving higher-level math problems.