Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, divide the rational expressions. \(\frac{9 x^{2}+3 x-20}{3 x^{2}-7 x+4} \div \frac{6 x^{2}+4 x-10}{x^{2}-2 x+1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{(3x+5)(x-1)}{2(3x-5)(x+1)}\).

Step 1: Understand the Division of Rational Expressions

To divide rational expressions, you multiply the first expression by the reciprocal of the second expression. If you have the expression \( A \div B \), it becomes \( A \times \frac{1}{B} \). Thus, the given expression becomes \( \frac{9x^2 + 3x - 20}{3x^2 - 7x + 4} \times \frac{x^2 - 2x + 1}{6x^2 + 4x - 10} \).

Step 2: Factorize Each Quadratic

Factor each quadratic expression. \(9x^2 + 3x - 20\) is factored as \((3x - 4)(3x + 5)\). \(3x^2 - 7x + 4\) is factored as \((3x - 4)(x - 1)\). \(6x^2 + 4x - 10\) is factored as \(2(3x^2 + 2x - 5)\), which is further simplified to \(2(3x - 5)(x + 1)\). \(x^2 - 2x + 1\) is factored as \((x - 1)^2\).

Step 3: Rewrite the Expression with Factorized Form

Substitute the factored forms into the expression: \[ \frac{(3x - 4)(3x + 5)}{(3x - 4)(x - 1)} \times \frac{(x - 1)^2}{2(3x - 5)(x + 1)} \].

Step 4: Cancel Common Factors

Identify and cancel the common factors in the numerator and denominator. Cancel \((3x - 4)\) and \((x - 1)\) from both sides: \[ \frac{(3x + 5)(x - 1)}{2(3x - 5)(x + 1)} \].

Step 5: Simplify the Remaining Expression

The simplified expression after canceling common terms is \(\frac{(3x + 5)(x - 1)}{2(3x - 5)(x + 1)}\). This is the final simplified form of the rational expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are mathematical expressions where the highest degree of the variable is squared. In general, these expressions take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.To better understand quadratic expressions, consider the expression \( 9x^2 + 3x - 20 \). Here, \( 9x^2 \) is the quadratic term because it involves \( x \) raised to the power of 2.

• The term \( 3x \) is the linear part of the expression.
• The number \( -20 \) is the constant term, which doesn't have an \( x \).

Understanding the components of quadratic expressions is crucial when performing operations such as factoring.

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions. To solve quadratic equations efficiently, we often need to factor them.For example, let's factor the quadratic expression \( 9x^2 + 3x - 20 \). We find two numbers that multiply to the product of the leading coefficient (9) and constant term (-20), and add to give the middle term (3).

• Rewrite the expression as \( (3x - 4)(3x + 5) \).
• These are the factors. \( (3x - 4) \) and \( (3x + 5) \) are multiplying to create the original expression.

By factoring expressions thoroughly, we prepare them for other operations, such as division or simplification.

Division of Expressions

Division of rational expressions involves multiplying by the reciprocal of the divisor expression. It's similar to division involving regular fractions.Given the expression \( \frac{9x^2 + 3x - 20}{3x^2 - 7x + 4} \div \frac{6x^2 + 4x - 10}{x^2 - 2x + 1} \), we change this division into multiplication:

• The reciprocal of \( \frac{6x^2 + 4x - 10}{x^2 - 2x + 1} \) is \( \frac{x^2 - 2x + 1}{6x^2 + 4x - 10} \).
• Therefore, we multiply the first expression by this reciprocal.

Writing the expressions in their factorized forms before multiplying helps to simplify the process further by easily identifying and canceling common factors.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. This often involves canceling out common factors present in both the numerator and the denominator.Once we have the product of two rational expressions, such as \( \frac{(3x - 4)(3x + 5)}{(3x - 4)(x - 1)} \times \frac{(x - 1)^2}{2(3x - 5)(x + 1)} \), we simplify it by:

• Canceling the common factor \( (3x - 4) \) from both numerator and denominator.
• Canceling \( (x - 1) \), which appears twice in the numerator, with one in the denominator.

After these cancellations, the expression simplifies to \( \frac{(3x + 5)(x - 1)}{2(3x - 5)(x + 1)} \). This is the simplest version we can achieve, having removed all redundant factors.