Problem 78 - Introductory Algebra for College Students

Question

Factor completely. $$-45 y^{3}-30 y^{2}-5 y$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \(5y (-9y^2 - 6y - 1)\).

Step 1: Identify common factors in the expression

Looking at our expression \(-45y^3 - 30y^2 - 5y\), it is clear that \(5y\) is a common term in each of the components.

Step 2: Factor out the common factor

We factor out the common factor - in this case, \(5y\), from each component in our expression. This leaves us with: \(5y (-9y^2 - 6y - 1)\)

Step 3: Simplify the expression within the brackets

Looking inside the bracket \(-9y^2 - 6y - 1\), we can't factor it further, so it remains as it is. Hence, our exercise is done here.

Key Concepts

Understanding the Common Factor

A common factor is a number or expression that divides each term of a polynomial without leaving any remainder. Identifying the common factor is the first step in simplifying a polynomial expression. In the math problem provided, we are given the polynomial expression \(-45y^3 - 30y^2 - 5y\). Each term in this expression includes the factor \(5y\).

• The term \(-45y^3\) is divisible by \(5y\) because \(-45 \div 5 = -9\) and \(y^3 \div y = y^2\), resulting in \(-9y^2\).
• For \(-30y^2\), \(-30 \div 5 = -6\) and \(y^2 \div y = y\), giving us \(-6y\).
• Finally, for \(-5y\), \(-5 \div 5 = -1\) and \(y \div y = 1\), which simplifies to \(-1\).

Recognizing the common factor \(5y\) helps you factor the expression more easily and is a crucial step in the process.

Exploring Polynomial Expressions

A polynomial expression consists of variables like \(y\) raised to various powers, and each term may include coefficients, which are numerical values. In the expression \(-45y^3 - 30y^2 - 5y\), we deal with a third-degree polynomial because the highest power of \(y\) is three.Here's what each component represents:

• \(-45y^3\): This term stands for "negative forty-five times \(y\) cubed."
• \(-30y^2\): This is "negative thirty times \(y\) squared."
• \(-5y\): "Negative five times \(y\)."

Polynomial expressions like this can be simplified using factoring techniques. By identifying common factors and other patterns, we can often reduce a complex polynomial to a simpler form that is easier to work with.

Factoring Completely

The process of factoring completely involves breaking down a polynomial expression into its simplest components. In our example, after factoring out the common factor \(5y\), we have:\[5y (-9y^2 - 6y - 1)\]Here, the original polynomial \(-45y^3 - 30y^2 - 5y\) was simplified by separating out \(5y\), which leaves us with \(-9y^2 - 6y - 1\).Factoring completely means ensuring that all parts of the polynomial are expressed in the simplest way possible. This usually involves:

• Identifying and factoring out the greatest common factor, as we did with \(5y\).
• Reassessing the polynomial within the brackets (or parentheses) to see if further factoring is possible. In our case, \(-9y^2 - 6y - 1\) cannot be simplified further.

This method is important for simplifying expressions and solving equations, making them easier to manage in both educational and real-world math applications. Each time you factor completely, you're bringing the expression down to its most accessible form.