Problem 12
Question
For the following exercises, simplify the rational expressions. \(\frac{3 c^{2}+25 c-18}{3 c^{2}-23 c+14}\)
Step-by-Step Solution
Verified Answer
\(\frac{c + 9}{c - 7}\) is the simplified form.
1Step 1: Identify the Numerator and Denominator
The given rational expression is \(\frac{3c^2 + 25c - 18}{3c^2 - 23c + 14}\). Here, the numerator is \(3c^2 + 25c - 18\) and the denominator is \(3c^2 - 23c + 14\).
2Step 2: Factor the Numerator
We need to factor the quadratic expression in the numerator, \(3c^2 + 25c - 18\). Look for two numbers that multiply to \(-54\) (the product of \(3\) and \(-18\)) and add to \(25\). These numbers are \(27\) and \(-2\). Rewrite the expression as \(3c^2 + 27c - 2c - 18\). Then, factor by grouping: \[3c(c + 9) - 2(c + 9) = (3c - 2)(c + 9)\].
3Step 3: Factor the Denominator
Factor the quadratic expression in the denominator, \(3c^2 - 23c + 14\). Look for two numbers that multiply to \(42\) (the product of \(3\) and \(14\)) and add to \(-23\). These numbers are \(-21\) and \(-2\). Rewrite the expression as \(3c^2 - 21c - 2c + 14\). Then, factor by grouping:\[3c(c - 7) - 2(c - 7) = (3c - 2)(c - 7)\].
4Step 4: Simplify the Expression
Now that both the numerator and denominator are factored, \(\frac{(3c - 2)(c + 9)}{(3c - 2)(c - 7)}\), cancel the common factor \(3c - 2\) from the numerator and the denominator. The simplified form is thus:\[c + 9\] divided by \[c - 7\].
Key Concepts
Quadratic ExpressionsFactoringSimplificationNumerator and Denominator
Quadratic Expressions
Quadratic expressions are fundamental in algebra and appear in the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. These expressions describe a parabolic curve when graphed. The degree of the expression is 2, which signifies the highest power of the variable.
In our exercise, both the numerator and the denominator of the rational expression are quadratics: \( 3c^2 + 25c - 18 \) and \( 3c^2 - 23c + 14 \). Understanding how to manipulate and factor these expressions is crucial for simplification of rational expressions. The key challenge is to rewrite them in a form that reveals any common factors.
In our exercise, both the numerator and the denominator of the rational expression are quadratics: \( 3c^2 + 25c - 18 \) and \( 3c^2 - 23c + 14 \). Understanding how to manipulate and factor these expressions is crucial for simplification of rational expressions. The key challenge is to rewrite them in a form that reveals any common factors.
Factoring
Factoring is the process of breaking down an expression into a product of simpler expressions that, when multiplied together, give the original expression.
In our exercise, we factor quadratic expressions by looking for two numbers that both add to the middle coefficient and multiply to the product of the leading coefficient and the constant term.
In our exercise, we factor quadratic expressions by looking for two numbers that both add to the middle coefficient and multiply to the product of the leading coefficient and the constant term.
- For \( 3c^2 + 25c - 18 \), we need numbers that multiply to \(-54\) and add up to \(25\), which are \(27\) and \(-2\).
- For \( 3c^2 - 23c + 14 \), we seek numbers that multiply to \(42\) and sum to \(-23\), which are \(-21\) and \(-2\).
Simplification
Simplifying a rational expression involves reducing it to its simplest form by canceling common factors found in both the numerator and the denominator.
After factoring both the numerator and denominator, our expression becomes\( \frac{(3c - 2)(c + 9)}{(3c - 2)(c - 7)} \)Here, the common factor \( (3c - 2) \) can be canceled from both the numerator and the denominator. This operation leaves us with the simplified expression, \( \frac{c + 9}{c - 7} \). The simplification process must ensure the expression remains equivalent and all valid restrictions (like division by zero) are considered.
After factoring both the numerator and denominator, our expression becomes\( \frac{(3c - 2)(c + 9)}{(3c - 2)(c - 7)} \)Here, the common factor \( (3c - 2) \) can be canceled from both the numerator and the denominator. This operation leaves us with the simplified expression, \( \frac{c + 9}{c - 7} \). The simplification process must ensure the expression remains equivalent and all valid restrictions (like division by zero) are considered.
Numerator and Denominator
The numerator and denominator are fundamental parts of any rational expression.
The numerator is the top part of the fraction, representing the dividend. In our example, initially, it is the quadratic expression \( 3c^2 + 25c - 18 \).
The denominator is the bottom part, representing the divisor. Initially, it is \( 3c^2 - 23c + 14 \). Each needs careful manipulation, especially during factoring and simplification, to extract common factors accurately.
The numerator is the top part of the fraction, representing the dividend. In our example, initially, it is the quadratic expression \( 3c^2 + 25c - 18 \).
The denominator is the bottom part, representing the divisor. Initially, it is \( 3c^2 - 23c + 14 \). Each needs careful manipulation, especially during factoring and simplification, to extract common factors accurately.
- Always identify these parts first in any rational expression.
- The canceling of common terms occurs between the numerator and denominator, simplifying the expression to its lowest terms.
- The denominator cannot be zero, as division by zero is undefined, influencing the domain of the expression.
Other exercises in this chapter
Problem 11
For the following exercises, simplify each expression. \(\sqrt{98}\)
View solution Problem 11
For the following exercises, simplify the given expression. Write answers with positive exponents. \(11^{3} \div 11^{4}\)
View solution Problem 12
For the following exercises, factor by grouping. \(6 c^{2}+41 c+63\)
View solution Problem 12
For the following exercises, find the sum or difference. \(\left(4 z^{3}+8 z^{2}-z\right)+\left(-2 z^{2}+z+6\right)\)
View solution