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Chapter 5: Problem 52

Write an equivalent expression by factoring. $$72 x^{3}-36 x^{2}+24 x$$

Short Answer

Expert verified
The equivalent expression by factoring is 12x(6x^2 - 3x + 2).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine the coefficients and the variables of each term in the expression to identify the greatest common factor (GCF). The terms in the expression are 72, 36, and 24 for the coefficients, and each term includes the variable x. The GCF of the coefficients 72, 36, and 24 is 12, and the GCF for the variable part is x.
02

Factor out the GCF

Divide each term in the expression by the GCF and write the expression as a product of the GCF and the resulting terms. The original expression is 72x^3 - 36x^2 + 24x. Factoring out the GCF 12x, we get: \[\begin{equation} \begin{aligned} &72 x^{3}-36 x^{2}+24 x \ &=12x(6x^{2}-3x+2) \ \text{Thus, the factored expression is } 12x(6x^{2}-3x+2). \ \ \textjug2] \end{aligned} \end{equation}\]. Note: \end{array}$ \for further clarity.
03

Result: Equivalent Factored Expression

The equivalent expression for the given expression 72x^3 - 36x^2 + 24x after factoring out 12x is 12x(6x^2 - 3x + 2).

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Key Concepts

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

Greatest Common Factor
To factor a polynomial expression, the first step is to identify the Greatest Common Factor (GCF). The GCF is the largest number and variable that each term in the polynomial shares. For the expression 72x^3 - 36x^2 + 24x, we need to find the GCF of the coefficients and the variables:
  • Look at the coefficients: 72, 36, and 24. The largest number that divides all three of these numbers is 12.
  • For the variable part: each term has at least one 'x'. Therefore, the GCF for the variable part is 'x'.

Hence, the GCF for the entire expression is 12x. Factoring out this GCF simplifies the polynomial, making it easier to work with.
Polynomial Expressions
A polynomial expression consists of terms that are either constants or variables raised to whole number powers, added or subtracted together. In our example, the polynomial is 72x^3 - 36x^2 + 24x.
Here are the key points to note:
  • Each term in the polynomial is a product of a coefficient and a variable.
  • The expression is a sum or difference of these terms.
  • The degree of the polynomial is determined by the term with the highest exponent, in this case, it's 3 (from 72x^3).
Understanding the structure of polynomial expressions is crucial. It helps in simplifying, factoring, and performing other algebraic operations on them.
Coefficients and Variables
Polynomials are composed of coefficients and variables. In 72x^3 - 36x^2 + 24x:
  • Coefficients are the numerical parts of the terms. They are 72, 36, and 24 in this example.
  • Variables are the symbolic parts, usually represented by letters like 'x', 'y', etc. Here, the variable is 'x'.
When factoring polynomials, both coefficients and variables are considered to find the GCF. After identifying the GCF and factoring it out, the terms inside the parentheses usually have simpler coefficients and variable expressions. For our given polynomial, the expression simplifies to 12x(6x^2 - 3x + 2) after factoring out 12x. This shows how understanding coefficients and variables helps in breaking down complex polynomial expressions.

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