Problem 88 For Exercises 78 to $131,$ fac... [FREE SOLUTION]

Chapter 10: Problem 88

For Exercises 78 to $131,$ factor completely. $$-3 y^{3}+15 y^{2}-18 y$$

Short Answer

Expert verified

The completely factored form of the polynomial $-3 y^{3}+15 y^{2}-18 y$ is $-3y(y-2)(y-3)$.

Step by step solution

Identify the Greatest Common Factor

In the given expression $-3 y^{3}+15 y^{2}-18 y$, the greatest common factor is -3y. This is the largest quantity that divides evenly into every term of the expression.

Factor out the GCF

We factor out the GCF by dividing each term by the identified GCF and write the polynomial as a product. In our case, we divide each term in the expression $-3 y^{3}+15 y^{2}-18 y$ by -3y. Doing so gives us:$-3 y^{3}/(-3y)+15 y^{2}/(-3y)-18 y/(-3y)$This can be simplified as:$-3y(y^{2}-5y+6)$

Break Down The Quadratic

Now, we have a quadratic expression $y^{2}-5y+6$ inside the bracket which can be factored further. We are looking for two numbers which add up to -5 (the coefficient of y) and multiply to 6 (the constant term). From inspection, we see that -2 and -3 satisfy this condition because (-2)+(-3) = -5 and (-2)*(-3) = 6. So, the quadratic can be broken down as follows:$-3y*[y-2][y-3]$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Greatest Common Factor

The greatest common factor (GCF) is an essential first step in factoring polynomials. It refers to the largest factor shared by all the terms in the polynomial. In our example, the polynomial $-3y^{3} + 15y^{2} - 18y$, each term can be divided evenly by -3y.

For $-3y^3$, dividing by -3y gives $y^2$.
For $15y^2$, dividing by -3y results in -5y.
For $-18y$, dividing by -3y yields 6.

By factoring out the GCF, the expression simplifies to $-3y(y^{2} - 5y + 6)$. Finding the GCF simplifies the polynomial, making the next steps easier and more manageable, which is crucial in higher level math.

Quadratic Factoring

Once you factor out the GCF from a polynomial expression, you may often be left with a quadratic expression to factor further. The quadratic $y^{2} - 5y + 6$ in our example is ready for factoring. Quadratic factoring involves finding two numbers that multiply to the constant term and add up to the linear coefficient. Here:

You need numbers that multiply to 6.
These numbers must also add to -5.

The numbers -2 and -3 meet these criteria since $(-2) \times (-3) = 6$ and $(-2) + (-3) = -5$. Thus, the factored form of the quadratic is $(y - 2)(y - 3)$. This process requires practice but is pivotal in simplifying polynomials to solve or further analyze them.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. The expression given, $-3y^{3} + 15y^{2} - 18y$, is a polynomial with three terms, making it a trinomial.
Polynomials are classified by:

The number of terms: monomial (one term), binomial (two terms), trinomial (three terms), or otherwise polynomial (more than three terms).
The degree: which is the highest power of the variable, in this case, 3.

Understanding polynomial structure is crucial in math because it influences how you approach solving or factoring them. Recognizing patterns, such as trinomial or binomial forms, aids in choosing appropriate factoring methods, simplifying calculations, and understanding broader algebraic concepts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

For Exercises 78 to \(131,\) factor completely. $$-3 y^{3}+15 y^{2}-18 y$$

Short Answer

Step by step solution

Identify the Greatest Common Factor

Factor out the GCF

Break Down The Quadratic

Key Concepts

Greatest Common Factor

Quadratic Factoring

Polynomial Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Probability and Statistics

Geometry

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.