Problem 63 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(6 y^{3}-8 y^{2}-30 y\)

Step-by-Step Solution

Answer

The factored form is \(2y(3y + 5)(y - 3)\).

Step 1: Identify the Common Factor

Look at all the terms in the trinomial: \(6y^3\), \(-8y^2\), and \(-30y\). Identify the greatest common factor (GCF) of the coefficients and the variable terms. Here, the GCF is \(2y\), as it is the largest factor that divides all coefficients and includes the lowest power of \(y\) present.

Step 2: Factor Out the GCF

Factor \(2y\) out of each term in the trinomial. Perform the division for each term as follows:\[2y(3y^2 - 4y - 15)\]

Step 3: Factor the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parenthesis: \(3y^2 - 4y - 15\). Look for two numbers whose product is \(3 \times (-15) = -45\) and sum is \(-4\). These numbers are \(5\) and \(-9\).

Step 4: Apply the Split Middle Term Method

Rewrite the middle term, \(-4y\), using \(5\) and \(-9\):\[3y^2 + 5y - 9y - 15\].Then, group the terms:\[(3y^2 + 5y) + (-9y - 15)\].

Step 5: Factor by Grouping

Factor out the common factors from each group:- From \((3y^2 + 5y)\), factor out \(y\), giving \(y(3y + 5)\).- From \((-9y - 15)\), factor out \(-3\), giving \(-3(3y + 5)\).The expression now is \[y(3y + 5) - 3(3y + 5)\].

Step 6: Factor the Common Binomial

Notice the common binomial \((3y+5)\) in both terms:\[(3y + 5)(y - 3)\].

Step 7: Combine the Factors

Finally, multiply the GCF \(2y\) by the factors from Step 6:\[2y(3y + 5)(y - 3)\].

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a fundamental concept in factoring polynomials. It refers to the largest factor that divides multiple terms or numbers completely without leaving a remainder. For polynomials, this involves both numerical coefficients and variables. Identifying the GCF is often the first step in solving a polynomial equation.

Let’s break it down with an easy example:

• Look at the polynomial terms: \(6y^3\), \(-8y^2\), and \(-30y\).
• Find the greatest numerical factor common to all the coefficients, which in this case is \(2\).
• Observe the variable terms; \(y^3\), \(y^2\), and \(y\) common in all have the lowest power of \(y\) that is \(y^1\).
• Combine to get the GCF \(2y\).

The purpose of factoring out the GCF is to simplify the polynomial expression and make it easier to work with. Once factored, the polynomial is written in a form where further factoring can take place if needed.

Trinomial Factoring

Trinomial factoring is the process of breaking down a polynomial with three terms (trinomial) into products of simpler polynomials. This is a common step in algebra particularly useful for simplifying expressions and solving quadratic equations.

Let's explore how you perform trinomial factoring:

• The quadratic expression from the example is \(3y^2 - 4y - 15\).
• First, identify two numbers that multiply to the product of the first and last coefficient \((3 \, \text{and} \, -15\), so \(3 \, \times \, -15 = -45\)).
• The same numbers should add up to the middle coefficient \(-4\). These numbers are \(5\) and \(-9\).
• Replace the middle term with these numbers: \(3y^2 + 5y - 9y - 15\).
• Now group the terms: \((3y^2 + 5y) + (-9y - 15)\).
• Factor each group: \(y(3y + 5) - 3(3y + 5)\).

The expression is now ready to be factored further by identifying common factors in each group, simplifying it to a product of binomials.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\). Factoring quadratic expressions is a significant part of solving quadratic equations, which are fundamental in algebra.

Here is a straightforward method for factoring quadratic expressions:

• Consider the quadratic expression \(3y^2 - 4y - 15\) as part of the polynomial \(6 y^{3}-8 y^{2}-30 y\).
• Identify the coefficients \(a = 3\), \(b = -4\), and \(c = -15\).
• Find two numbers that multiply to the product \(a \times c = -45\) and add up to \(b = -4\). The numbers are \(5\) and \(-9\).
• Rewrite the quadratic using these values to split the middle term: \(3y^2 + 5y - 9y - 15\).
• Grouping the terms makes it possible to factor: \(y(3y + 5) - 3(3y + 5)\).
• Finally, factor out the common binomial pair: \((3y + 5)(y - 3)\).

This leads to the complete factored form of the quadratic expression, making it straightforward to solve or simplify further if needed.