Problem 51 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. $$ 3 y^{3}+15 y^{2}-18 y $$

Step-by-Step Solution

Answer

The factored expression is \(3y(y-1)(y+6)

Step 1: Identify the common factor

The expression is \(3y^3 + 15y^2 - 18y). From the coefficients and variables in each term, it can be seen that 3y is a common factor for all three terms.

Step 2: Factor out the common factor

Factor out 3y from the expression which will give \(3y(y^2 + 5y - 6)\).

Step 3: Factor the resulting polynomial

The quadratic equation in the brackets \(y^2 + 5y - 6\) can be factored further into \((y-1)(y+6)\).

Step 4: Put it all together

Combine the common factor and the factored polynomial resulting in the completely factored expression of the original: \(3y(y-1)(y+6)\)

Key Concepts

Understanding the Greatest Common Factor

When working with polynomials, finding the greatest common factor (GCF) is an essential first step in factoring. The GCF is the largest term that divides all the terms in the polynomial evenly. For the expression \(3y^3 + 15y^2 - 18y\), we look at all the coefficients and variables. The coefficients are 3, 15, and 18, and they all can be divided by 3. Additionally, each term contains the variable \(y\), with the smallest power being \(y^1\).

To find the GCF, follow these steps:

• Look at the coefficients of each term. Find the largest number that divides them all. Here, it's 3.
• Look at the variables in each term. The smallest power of \(y\) that is common in all terms is \(y\).

Thus, the GCF of \(3y^3 + 15y^2 - 18y\) is \(3y\). This process simplifies the polynomial and is a necessary step before further factoring.

Exploring Quadratic Equations

Quadratic equations are a special type of polynomial that can be written in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, after factoring out \(3y\) from the expression, we are left with a quadratic equation inside the brackets—\(y^2 + 5y - 6\).

Factoring quadratic equations involves finding two binomials that multiply together to give the original equation. For \(y^2 + 5y - 6\):

• Look for two numbers that multiply to \(-6\) (the constant term \(c\)) and add to \(5\) (the middle coefficient \(b\)).
• The numbers \(-1\) and \(6\) fit these conditions, as \((-1) \times 6 = -6\) and \((-1) + 6 = 5\).

Thus, \(y^2 + 5y - 6\) can be factored into \((y - 1)(y + 6)\). This is an essential technique for breaking down and solving quadratic equations.

Breaking Down Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations, and the goal of factoring is to simplify or re-write them in a more usable form. Take \(3y^3 + 15y^2 - 18y\) for example, where the expression involves multiple terms with both numerical coefficients and variables.

By applying the concept of the GCF and further factoring, we rewrite it as \(3y(y-1)(y+6)\). Here's how it helps:

• The original expression is simplified into a product of simpler expressions.
• It shows clearly each mathematical operation performed on the variable \(y\).
• This form makes it easier to solve equations or analyze behavior when set equal to zero.

Factoring is a crucial skill when dealing with algebraic expressions, making complex problems more manageable and solving equations more straightforward.