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Chapter 1: Problem 97

Factor the expression completely. $$12 x^{3}+18 x$$

Short Answer

Expert verified
The completely factored expression is \(6x(2x^2 + 3)\).

Step by step solution

01

Identify Common Factors

Look at each term in the expression \(12x^3 + 18x\). Both terms have a numerical coefficient and a variable part. Determine the greatest common factor (GCF) of the coefficients 12 and 18, which is 6. Additionally, notice that both terms include the variable \(x\). The highest power of \(x\) that is a factor of both terms is \(x\) itself. Hence, the GCF of the entire expression is \(6x\).
02

Factor Out the GCF

Divide each term by the GCF \(6x\) to factor the expression: \[ \frac{12x^3}{6x} = 2x^2 \] and \[ \frac{18x}{6x} = 3 \]. Therefore, \(12x^3 + 18x = 6x(2x^2 + 3)\).
03

Verify the Factored Expression

To ensure the expression is factored correctly, distribute \(6x\) back into \((2x^2 + 3)\) to see if the original expression is retrieved: \[ 6x(2x^2) + 6x(3) = 12x^3 + 18x \]. This matches the original expression, confirming the factorization is correct.

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Key Concepts

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

Greatest Common Factor
When dealing with polynomial expressions, locating the Greatest Common Factor (GCF) is a crucial step in simplifying expressions. The GCF is the highest factor that can evenly divide each term of the polynomial. To find the GCF:
  • Identify the coefficients of each term; in our example, we have 12 and 18.
  • Find the largest number that divides both coefficients without leaving a remainder. For 12 and 18, this number is 6.
  • Examine the variable parts. Here, both terms have the variable \(x\). The shared factor, in this case, is the lowest power of \(x\), which is \(x^1\) or simply \(x\).

The GCF for \(12x^3 + 18x\) becomes \(6x\). Factoring out the GCF is like finding common parts that can be grouped together to simplify the problem.
Factoring Techniques
Factoring techniques allow you to express a polynomial in simpler terms. Here's how to factor using the GCF method:
  • After identifying the GCF, divide each term in the polynomial by this factor. This is like "unwrapping" each term, revealing simpler components.
  • In the example of \(12x^3 + 18x\), factor out \(6x\) from each term:
\[\frac{12x^3}{6x} = 2x^2\] and \[\frac{18x}{6x} = 3\].
Thus, the original expression \(12x^3 + 18x\) becomes \(6x(2x^2 + 3)\). Factoring relies on finding patterns and grouping terms for easier manipulation, simplifying processes in solving equations or analyzing polynomials.
Polynomial Expressions
Polynomial expressions are algebraic expressions that involve variables and coefficients, composed of terms combined through addition and subtraction. A polynomial can have constants, variables, and non-negative integer exponents.
  • Each term in a polynomial is defined by a coefficient and a variable raised to an exponent.
  • The term structure, such as \(12x^3\) and \(18x\), can vary in complexity, from simple monomials to complex polynomials.
  • The degree of a polynomial indicates the highest power of the variable within that expression.

Factoring polynomials, like \(12x^3 + 18x\), involves breaking down the expression into product forms. This process is vital in algebra for simplifying expressions and solving polynomial equations, as it underscores the interplay between numerical coefficients and variable elements.

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Most popular questions from this chapter

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