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Chapter 7: Problem 62

Factor: $$6 x^{3}-6 x^{2}-120 x$$

Short Answer

Expert verified
The factored form of the given expression is \(6x(x-5)(x+4)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

In this case, the GCF of \(6x^{3}\), \(-6x^{2}\), and \(-120x\) is \(6x\).
02

Factor out the GCF

We factor out \(6x\) from each term which leaves us with \(6x(x^{2} - x - 20)\).
03

Factor the quadratic

Now, factor the quadratic \(x^{2}- x - 20\). This can be broken down into \((x - 5)(x + 4)\), so now the fully factored expression is \(6x(x-5)(x+4)\).

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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 number that can evenly divide two or more numbers. When working with algebraic expressions, the GCF also considers the variable part of the terms.
To find the GCF of terms like \(6x^3\), \(-6x^2\), and \(-120x\), follow these steps:
  • First, look at the coefficients (numbers in front of the variables). Here, the coefficients are 6, -6, and -120. The GCF of these numbers can be found by looking for the largest integer that divides all of them. This is 6.
  • Next, consider the variable parts. Since all terms have at least one \(x\), the GCF of the variables is \(x\).

Combining both aspects, the GCF of the entire expression \(6x^3 - 6x^2 - 120x\) is \(6x\). Finding the GCF is a vital step in simplifying and factoring algebraic expressions as it sets the stage for the further breaking down of the expression.
Quadratic Equation
A quadratic equation is any polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown. The expression \(x^2 - x - 20\) is an example of a quadratic, which is often the result of factoring algebraic expressions.
To factor the quadratic \(x^2 - x - 20\):
  • Look for two numbers that multiply to \(-20\) (the constant term) and add to \(-1\) (the coefficient of the \(x\) term).
  • These numbers are \(-5\) and \(4\), which means we can rewrite the quadratic as \((x - 5)(x + 4)\).

Factoring quadratics is essential since it allows you to solve equations and simplify expressions. Always check your work by expanding the factors to ensure they match the original quadratic.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition or multiplication). Understanding how to simplify these expressions is crucial for solving equations and tackling more complex math problems.
Simplifying involves reducing the expression to its smallest form, often by factoring. Here, the expression \(6x(x^2 - x - 20)\) was simplified further:
  • By identifying the GCF and factoring it out, we reduced the complexity of the expression.
  • We factored \(x^2 - x - 20\) into \((x - 5)(x + 4)\), allowing a clearer view of the expression's structure.
  • Thus, the complete factored form becomes \(6x(x - 5)(x + 4)\).

Factoring and simplifying algebraic expressions not only streamlines calculations but also opens doors to deeper analysis and understanding of mathematical behaviors and relationships.

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Most popular questions from this chapter

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3}{x^{2}+4 x y+3 y^{2}}-\frac{5}{x^{2}-2 x y-3 y^{2}}+\frac{2}{x^{2}-9 y^{2}}$$

Exercises \(100-102\) will help you prepare for the material covered in the next section. If you can complete a job in 5 hours, what fractional part of the job can you complete in 1 hour? in 3 hours? in \(x\) hours?

Multiply: \(\quad(3 x+5)(2 x-7)\).

Divide: $$\frac{27 x^{3}-8}{3 x+2}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}+2 y+1}+\frac{4}{y^{2}+5 y+4}$$
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