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Chapter 5: Problem 68

Factor completely. $$ -9 x^{3}+12 x $$

Short Answer

Expert verified
The completely factored form is \(3x(-3x^2 + 4)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the polynomial \(-9x^3 + 12x\). Both terms have a common factor of \(3x\).
02

Factor Out the GCF

Factor out the GCF \(3x\) from each term: \(-9x^3 + 12x = 3x(-3x^2 + 4)\).
03

Check for Further Factoring

Examine the remaining polynomial \(-3x^2 + 4\). There are no further common factors and it cannot be factored further into simple polynomials.
04

Write the Final Answer

Conclude that the completely factored form of \(-9x^3 + 12x\) is \3x(-3x^2 + 4)\.

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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 the largest factor that divides two or more numbers. In algebra, it's used to simplify polynomials and can make complex equations more manageable. To find the GCF of polynomial terms, we look for the highest number and variable exponent that each term has in common. Let's consider the problem: \(-9x^3 + 12x\). Both terms, \-9x^3 \and \ 12x\, share a common factor. Here, the number 3 and the variable \ \( x \) \ appear in both terms. Therefore, the GCF is \ \( 3x \). Finding the GCF is the first and crucial step before factoring polynomials.
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients. They involve operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For instance, \-9x^3 + 12x\ is a polynomial. Each term of the polynomial is composed of a coefficient (like -9 or 12) and a variable raised to an exponent (like x^3 or x). When simplifying polynomials, we often factor them to find simpler forms. Factoring involves expressing the polynomial as a product of its factors. This can reveal useful properties about the equation and make solving them easier. In our exercise, we've shown how to factor \-9x^3 + 12x \ into \ \(3x(-3x^2 + 4)\).
Intermediate Algebra
Intermediate algebra builds upon basic algebra principles and extends into more complex topics. These topics include factoring, solving quadratic equations, and dealing with polynomials and rational expressions. In our example, factoring the polynomial involves using fundamental concepts of intermediate algebra like finding the GCF and rewriting polynomials in simpler forms. Steps include recognizing common factors and ensuring no further factoring is possible. Mastering intermediate algebra provides essential skills needed for advanced mathematical problem-solving, and is a stepping stone for calculus and higher math courses.

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