Problem 67 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. $$\frac{3 x^{2}+3 x-60}{2 x-8}+\left(\frac{30 x^{2}}{x^{2}-7 x+10} \cdot \frac{x^{3}+3 x^{2}-10 x}{25 x^{3}}\right)$$

Step-by-Step Solution

Answer

After performing the given operations, the resulting algebraic fraction becomes \(\frac{3/2(x+5)(x-2)(x-5)25+30x(x^{2}+3 x-10)}{(x-2)(x-5)25}, which could be further simplified by distributing the components in the numerator.

Step 1: Factorization

Factorize the polynomials in the fraction \(\frac{3 x^{2}+3 x-60}{2 x-8}\) to get \(\frac{3(x+5)(x-4)}{2(x-4)}\). Also, in \(\frac{30 x^{2}}{x^{2}-7 x+10}\), the denominator can be factorized as (x-2)(x-5). Therefore the fraction becomes \(\frac{30x^2}{(x-2)(x-5)}\). Further, the \(x^{3}\) term in \(\frac{x^{3}+3 x^{2}-10 x}{25 x^{3}}\) can be taken common to give \(x(\frac{x^{2}+3 x-10}{25x^{2}})\).

Step 2: Simplification

Next, further simplify the fractions. \(\frac{3(x+5)(x-4)}{2(x-4)}\) becomes \(\frac{3}{2}(x+5)\) because the term (x-4) in the numerator and the denominator cancel out each other. Moreover, the \(x\) term in \(x(\frac{x^{2}+3 x-10}{25x^{2}})\) gets canceled to give \(\frac{(x^{2}+3 x-10)}{25x}\).

Step 3: Multiplication of fractions

Now, it is time to multiply the fraction \(\frac{30x^2}{(x-2)(x-5)}\) with \(\frac{(x^{2}+3 x-10)}{25x}\). The multiplication expression becomes \(\frac{30x^2(x^{2}+3 x-10)}{(x-2)(x-5)25x}\). Here, \(\frac{30x^2}{x}\) gets simplified to \(\frac{30x}{1}\). Thus, the multiplication result simplifies to \(\frac{30x(x^{2}+3 x-10)}{(x-2)(x-5)25}\).

Step 4: Addition of fractions

Now, add the two major compartments together. + \(\frac{3}{2}(x+5)\) to + \(\frac{30x(x^{2}+3 x-10)}{(x-2)(x-5)25}\). This requires a common denominator which is (x-2)(x-5)25. Therefore, the final expression is \(\frac{3/2(x+5)(x-2)(x-5)25+30x(x^{2}+3 x-10)}{(x-2)(x-5)25}\). The numerator can further be simplified using the distributive law.

Step 5: Simplification

Simplify the final answer by applying distributive law to \(\frac{3/2(x+5)(x-2)(x-5)25+30x(x^{2}+3 x-10)}{(x-2)(x-5)25}.\)

Key Concepts

Factorization

Factorization is an essential step in simplifying algebraic fractions. This technique involves breaking down a complex polynomial into its simpler components. For instance, if you have the expression \( 3x^2 + 3x - 60 \), you can factor it by looking for two numbers that multiply to make \(-60\) and add to make \(3\). Here, you get \((x+5)(x-4)\), where \((x+5)\) and \((x-4)\) are the factors.
Another example is the fraction \( \frac{30x^2}{x^2-7x+10} \). Factor the denominator to \( (x-2)(x-5) \). This step is crucial because it simplifies the expression and makes it easier to identify any components that can cancel each other out.
Remember, the essence of factorization is transforming the polynomial into a product of simpler factors. This approach reduces complexity and helps in carrying out further operations like simplification and multiplication easily.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by canceling common factors in the numerator and the denominator of fractions. This simplification is key to making calculations easier and more manageable.
Take the expression \( \frac{3(x+5)(x-4)}{2(x-4)} \) as an example. Notice that \((x-4)\) appears in both the numerator and denominator, allowing you to cancel it out and simplify the fraction to \( \frac{3}{2}(x+5) \).
Similarly, in the expression \( x(\frac{x^2 + 3x - 10}{25x^2}) \), you can cancel the \( x \) in the numerator with one in the denominator to simplify it to \( \frac{x^2 + 3x - 10}{25x} \).
Always look for opportunities to cancel out terms in both the numerator and the denominator. This practice not only helps in simplifying the fraction but also sets the stage for efficient multiplicative operations.

Polynomial Operations

Polynomial operations encompass addition, subtraction, multiplication, and division of polynomial expressions. When dealing with fractions involving polynomials, after simplification comes multiplication and addition.
The first operation is multiplication. Consider the fraction \( \frac{30x^2}{(x-2)(x-5)} \) being multiplied by \( \frac{x^2+3x-10}{25x} \). First, simplify the expression \( \frac{30x^2}{x} \) to \( 30x \) by canceling an \(x\). Then multiply the numerators and the denominators: \( \frac{30x(x^2+3x-10)}{(x-2)(x-5)25} \).
The next relevant operation is addition. When summing the fractions \( \frac{3}{2}(x+5) \) and \( \frac{30x(x^2+3x-10)}{(x-2)(x-5)25} \), convert them to a common denominator. In this case, the least common denominator is \((x-2)(x-5)25\). Then, add the two expressions to form one complete expression.
Remember to simplify the final result whenever possible to achieve the simplest form of the expression.