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Chapter 11: Problem 12

Simplify. $$\frac{12 x^{2}(3-x)}{18 x(3-x)}$$

Short Answer

Expert verified
\(\frac{2}{3}x\). This is our simplified version of the original fraction.

Step by step solution

01

Simplify the coefficients

Looking at the coefficients, we have \(12/18\). Both numbers are divisible by 6, so simplifying this fraction gives us \(2/3\). Thus, the fraction becomes \(\frac{2}{3}\cdot \frac{x^{2}(3-x)}{x(3-x)}\) .
02

Simplify the x terms

In the numerator we have \(x^{2}\) and in the denominator we have \(x\). If you divide \(x^{2}\) by \(x\), you are left with \(x\). As a result, the fraction becomes \(\frac{2}{3}x\).
03

Cancel out identical terms

We have the term \((3-x)\) both at the numerator and the denominator. Since they are identical, they can be cancelled out. After cancelling, we are left with: \(\frac{2}{3}x\). This is our simplified expression.

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Key Concepts

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

Canceling Terms
Canceling terms is a fundamental concept in algebra that simplifies expressions by removing identical factors from both the numerator and the denominator. When an expression contains common factors, these can be canceled, as long as they are not equal to zero. For example, if you have the algebraic fraction \( \frac{a(b-c)}{d(b-c)} \), and \(b-c eq 0\), you can cancel out \( (b-c) \) from both parts:
  • Numerator: The top part of the fraction, where terms can be canceled if they match the denominator.
  • Denominator: The bottom part of the fraction, it needs matching terms from the numerator to simplify.
  • Important: Only cancel terms if they appear exactly the same and do not equal zero.
Canceling terms can greatly simplify algebraic expressions and make solving problems more efficient. Just remember to check that the canceled term does not evaluate to zero, as dividing by zero is undefined.
Simplifying Coefficients
In algebra, simplifying coefficients involves reducing numerical fractions by dividing the numerator and the denominator by their greatest common divisor (GCD). This process makes the expression more straightforward. For example, in the expression \( \frac{12}{18} \), both the numerator and the denominator can be divided by 6, their GCD, which reduces the fraction to \( \frac{2}{3} \).
  • Find the GCD: Identify the largest number that divides both the numerator and the denominator without a remainder.
  • Divide: Once the GCD is found, divide both the numerator and the denominator by this number.
  • Simplification: The result is a simpler form of the original fraction, easier to work with.
Remember, simplifying coefficients helps make the equation or expression easier to manage and solve, especially when involving further operations in algebraic expressions.
Algebraic Fractions
Algebraic fractions are expressions where the numerator and/or the denominator includes algebraic terms. Unlike regular fractions, which consist of numbers only, algebraic fractions involve variables, which can make them seem complex at first. A typical algebraic fraction might look like \( \frac{ax + b}{cx + d} \).
  • Numerator: Can include constants and variable terms.
  • Denominator: Similarly, it can have constants and variable terms, but very importantly, it cannot be zero.
  • Simplification: Involves reducing equations by canceling out common factors in numerator and denominator.
When working with algebraic fractions, the goal is to simplify by canceling terms and reducing coefficients, ultimately making them easier to handle in broader calculations.

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