Problem 26 - Elementary Algebra

Question

Given the GCF, determine the missing factor. $$ 30 y 3-45 y 2-3 y=3 y(\quad ? \quad) $$

Step-by-Step Solution

Answer

The missing factor is \(10y^2 - 15y - 1\).

Step 1: Factor Out the GCF

The expression is given as \(30y^3 - 45y^2 - 3y\). We need to factor out the greatest common factor (GCF) which is \(3y\) in this case. Dividing each term by \(3y\), we have:\[\frac{30y^3}{3y} = 10y^2, \quad \frac{45y^2}{3y} = 15y, \quad \text{and} \quad \frac{3y}{3y} = 1.\] Therefore, the expression inside the parenthesis is \(10y^2 - 15y - 1\).

Step 2: Write the Expression Inside Parentheses

After factoring out the GCF, the complete expression can be written as \(3y(10y^2 - 15y - 1)\). This shows that the missing factor, once the \(3y\) is factored out, is \(10y^2 - 15y - 1\).

Step 3: Verify the Factorization

Multiply back the factored form to check if it equals to the original expression. \[3y(10y^2 - 15y - 1) = (3y \cdot 10y^2) - (3y \cdot 15y) - (3y \cdot 1) = 30y^3 - 45y^2 - 3y.\] The multiplication confirms our factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of a large expression. It involves breaking down a complex polynomial into simpler terms multiplied together, much like how you might decompose numbers into prime factors. In our given exercise, we deal with the polynomial \(30y^3 - 45y^2 - 3y\). To factor it, we first identify the greatest common factor (GCF) among the terms. Here, it is \(3y\). By dividing each part of the polynomial by \(3y\), we simplify the expression. Essentially, this process reduces the polynomial to its essence, which is crucial for simplifying algebraic expressions further. Factoring helps in solving equations, analyzing polynomial behavior, or simplifying fractions.

Algebraic Expressions

An algebraic expression combines numbers, variables, and operations. Think of it as a mathematical phrase. The expression in our exercise, \(30y^3 - 45y^2 - 3y\), uses variables and numerical coefficients. Algebraic expressions, unlike equations, don’t have an equality sign. They are instead simplified or factored for different purposes. Here, finding the GCF and then factoring part of the expression helps us simplify it. This simplification makes it easier to handle, especially when dealing with more complicated algebra problems. Understanding these expressions is foundational in algebra as they represent real-world quantities or abstract concepts.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. In the context of our practice problem, it involves dividing each term of \(30y^3 - 45y^2 - 3y\) by the GCF, which is \(3y\). This step serves as a backbone for the process of factoring the polynomial. By breaking down each term, we identify the simplified terms \(10y^2 - 15y - 1\). Polynomial division not only aids in simplifying expressions but is also pivotal when finding roots, analyzing polynomial behavior, or performing algebraic long division. It’s a key concept that reveals the underlying structure of the polynomial.