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Chapter 8: Problem 42

Write each rational expression in lowest terms. $$\frac{3 c^{2}-36 c+96}{c-8}$$

Short Answer

Expert verified
The simplified rational expression in lowest terms is \(3(c - 4)\).

Step by step solution

01

Factor the numerator

First, we need to factor the numerator of the given expression: \(\frac{3 c^{2}-36 c+96}{c-8}\) We notice a common factor of 3 among all terms in the numerator, so let's factor it out: \(\frac{3(c^2 - 12c + 32)}{c-8}\) Next, let's try factoring the quadratic expression inside the parentheses: \(c^2 - 12c + 32\) We are looking for two numbers that multiply to 32 and add to -12. These numbers are -4 and -8. So, we can factor the quadratic expression as: \((c - 4)(c - 8)\) Now, let's substitute the factored expression back into the given rational equation: \(\frac{3(c - 4)(c - 8)}{c - 8}\)
02

Cancel out common factors

Now we can cancel out any common factors between the numerator and the denominator. In this case, we can see that both the numerator and the denominator have a factor of (c - 8): \(\frac{3(c - 4)\cancel{(c - 8)}}{\cancel{(c-8)}}\)
03

Write the simplified expression

After canceling out the common factor, we can write the simplified expression as: \(\frac{3(c - 4)}{1}\) Or simply: \(3(c - 4)\) So the rational expression in lowest terms is: \(3(c - 4)\)

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a vital skill when dealing with algebraic expressions, especially rational expressions. When given a quadratic expression such as \(c^2 - 12c + 32\), it is important to identify two numbers that multiply together to give the constant term (32 in this case) and add up to give the coefficient of the linear term (-12).
  • The numbers -4 and -8 meet these criteria, since \(-4 + (-8) = -12\) and \(-4 \times -8 = 32\).
  • Thus, the original quadratic expression can be rewritten as \((c - 4)(c - 8)\). This form is useful because it allows us to easily cancel common factors when simplifying rational expressions.
The factoring process not only makes complex algebraic problems more manageable but is a crucial step before simplifying expressions.
Canceling Common Factors
Once we factor both the numerator and the denominator in a rational expression, the next step involves canceling any common factors they share. In our example, the rational expression is\(\frac{3(c - 4)(c - 8)}{c - 8}\).You can see that both the numerator and the denominator share a common factor, \((c - 8)\).
Canceling common factors means removing the same terms from the numerator and the denominator, hence simplifying the expression.
  • Identify the common factor, in this case, \((c - 8)\).
  • Cross out this factor from both the numerator and the denominator. This simplifies the expression to \(3(c - 4)\).
This step is crucial as it ensures that the expression is in its simplest form, allowing for easier interpretation and use.
Rational Expressions
Rational expressions are akin to fractions but involve polynomials in their numerators and denominators. To simplify these expressions, one must
  • First, factor both the numerator and the denominator as completely as possible.
  • Then, cancel out any common factors between them.
In simplifying \(\frac{3 c^{2}-36 c+96}{c-8}\), our final result was \(3(c - 4)\).This process of simplification is an essential skill in algebra, especially for expressions involving multiple terms. It helps reveal the most reduced form of a problem, providing clearer insight into its structure and behavior. Overall, mastering rational expressions benefits greatly from an ability to deftly factor and cancel as necessary.

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Most popular questions from this chapter

Solve each equation. $$\frac{9}{4 x+1}=\frac{3}{x+2}$$

Perform the indicated operations. $$\frac{y+6}{y^{2}-4 y}+\frac{y}{2 y^{2}-13 y+20}-\frac{1}{2 y^{2}-5 y}$$

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Solve each equation. $$\frac{r-2}{2}=\frac{3 r-5}{4}$$

Perform the indicated operations. $$\frac{g-5}{5 g^{2}-30 g}+\frac{g}{2 g^{2}-17 g+30}-\frac{6}{2 g^{2}-5 g}$$
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