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Chapter 6: Problem 72

Factor completely. $$10 y^{3}+12 y^{2}+2 y$$

Short Answer

Expert verified
The complete factor form of the equation \(10y^3 + 12y^2 + 2y\) is \(2y(5y^2 + 6y + 1)\).

Step by step solution

01

Identify the Greatest Common Factor

Identify the greatest common factor of \(10y^3\), \(12y^2\) and \( 2y \). Here the GCF is \(2y\).
02

Factor out the GCF

Now, we factor this GCF, \(2y\) out from each term. So, it becomes \(2y(5y^2 + 6y + 1)\).
03

Check for factorable Quadratic

Now, we check if the quadratic expression in the bracket can be factored. But \(5y^2 + 6y + 1\) does not factor further.

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Key Concepts

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

Greatest Common Factor (GCF)
Finding the Greatest Common Factor (GCF) is the first step in factoring polynomials. It refers to the largest factor that divides each term of the polynomial without leaving a remainder. When faced with a polynomial problem, determining the GCF helps in simplifying the expression.
Consider the polynomial given:
  • For the terms: 10, 12, and 2, we first consider the greatest number that can evenly divide them all. This number is 2.
  • Next, focus on the variables, looking at the smallest power among them. Here, you'll notice that each term includes 'y'. The smallest power of 'y' in the terms is 'y', which aligns with the variable part of the GCF.
Combining these, the GCF of the expression is 2y. Identifying the GCF is crucial as it sets the stage for further simplification.
Quadratic Expression
A quadratic expression in the context of polynomial factoring is a polynomial of the form \(ax^2 + bx + c\). These expressions are incredibly common and the subject of multiple factoring methods due to their frequent appearance in mathematics.
In the exercise, after extracting the GCF, we are left with another expression:
  • In our case, this is \(5y^2 + 6y + 1\).
  • This is a classic quadratic expression where the coefficients are: \(a = 5\), \(b = 6\), and \(c = 1\).
When factoring quadratic expressions, the goal is to express it as a product of two binomials. However, some quadratics, like \(5y^2 + 6y + 1\), might not factor neatly into binomials using integers, hence they remain in their current form if no feasible factorization is possible.
Polynomial Factoring Steps
Factoring a polynomial mainly consists of a systematic process. Let's go through the general steps which should always be your guide:
  • **Step 1:** Always start by finding the Greatest Common Factor (GCF) of the polynomial terms. This simplifies the polynomial and makes it easier to manage.
  • **Step 2:** Factor out the GCF from all terms in the polynomial, leaving you with a cleaner, reduced expression. This yields the next segment of the polynomial for examination.
  • **Step 3:** Investigate further if the remaining expression can be simplified further. This could involve applying special methods depending on the polynomial type, such as factoring quadratics into binomials.
In our given problem, after factoring out the GCF of 2y, we examined \(5y^2 + 6y + 1\) for further factoring. However, finding no integers that work, we concluded our process there. Following these factoring steps improves understanding and increases chances of correct solutions.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because some trinomials are prime, some quadratic equations cannot be solved by factoring.

Solve: \(5 x+28=6-6 x\). (Section \(2.2,\) Example 7 )

In Exercises \(130-133,\) use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-9=(4 x+3)(4 x-3)$$

In Exercises \(137-141,\) factor completely. $$(x+5)^{2}-20(x+5)+100$$

Solve equation and check your solutions. \(y^{3}+3 y^{2}+2 y=0\)
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