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Chapter 5: Problem 28

Factor each trinomial. See Examples 5 through 10. $$ 18 y^{3}+6 y^{2}+2 y $$

Short Answer

Expert verified
The trinomial factors as \(2y(9y^2 + 3y + 1)\).

Step by step solution

01

Identify a Common Factor

First, we look for the greatest common factor (GCF) of all the terms. In the trinomial \(18y^3 + 6y^2 + 2y\), the coefficients are 18, 6, and 2. The GCF of these coefficients is 2. Additionally, each term has at least one \(y\) as a factor, so we have \(y\) as a common factor as well. Therefore, the GCF of the entire trinomial is \(2y\).
02

Factor Out the GCF

We factor out \(2y\) from each term in the trinomial:\[ 18y^3 + 6y^2 + 2y = 2y(9y^2 + 3y + 1) \]
03

Check the Factored Expression

The expression inside the parentheses, \(9y^2 + 3y + 1\), is a quadratic trinomial. To ensure the expression is fully factored, we check if \(9y^2 + 3y + 1\) can be factored further. In this case, the trinomial does not have two binomials that multiply to it, meaning it is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is a crucial element in factoring. It refers to the largest factor that divides all the terms of an expression evenly. When working with a trinomial like \(18y^3 + 6y^2 + 2y\), our first task is identifying the GCF among the numbers and variables present.
Here is how to determine it step by step:
  • Look at the coefficients: 18, 6, and 2. The largest number that divides all three is 2.
  • For variables, check the power of \(y\) in each term: \(y^3, y^2,\) and \(y\). The smallest power is \(y^1\). Therefore, \(y\) is a part of the GCF.
The GCF here is \(2y\). By factoring out the GCF, the trinomial reduces to a simpler form, making further factorization easier.
Quadratic Trinomial
A quadratic trinomial is a type of polynomial that contains three terms with the highest power of the variable being two. In our example, the expression \(9y^2 + 3y + 1\) inside the parentheses is a quadratic trinomial. It takes the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
Quadratic trinomials are crucial in algebra, appearing in a wide array of problems. To factor these further, we usually look for the following:
  • Two numbers that multiply to give the product of \(a\) and \(c\), and add up to \(b\).
  • If such numbers exist, the trinomial can be expressed as a product of two binomials.
  • If they don't exist, as in our case here, the trinomial is already in its simplest form.
Understanding the structure and behavior of quadratic trinomials is key to mastering the factoring process.
Factoring Process
The factoring process is a strategic breakdown of mathematical expressions. It involves identifying and extracting common factors or recognizing patterns that simplify the expression into more manageable parts. Here's a streamlined approach:
  • Start by identifying the GCF of the entire expression to simplify the terms as much as possible.
  • Look at what's left inside the parentheses; if it's a quadratic trinomial, check for patterns that allow further factoring.
  • Continue verifying each step by expanding your factored terms to see if you arrive back at the original expression, ensuring accuracy.
This process transforms complex polynomial expressions into simpler, workable forms. It's especially handy in solving equations and making sense of algebraic expressions. As you get more accustomed to this process, it becomes an indispensable part of tackling algebra problems efficiently.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ 3 x^{2}+6 x-45 $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ x^{2}+c x+16 $$

The volume \(V(x)\) of a box in terms of its height \(x\) is given by the function \(V(x)=x^{3}+2 x^{2}-8 x .\) Factor this expression for \(V(x)\). (GRAPH CANNOT COPY)

Factor. Assume that variables used as exponents represent positive integers. $$ 25 x^{2 n}-81 $$

Factor each trinomial. See Examples 5 through 10. $$ 28 y^{2}+22 y+4 $$
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