Problem 22 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{45 x^{3}-9 x^{2}}{x} \cdot \frac{2}{6(x-5)}$$

Step-by-Step Solution

Answer

The product in simplest form is \(15x - 3\). Remember that this solution is valid only for \(x ≠ 0\) and \(x ≠ 5\).

Step 1: Factor out the common factors

Firstly, you want to factor out each expression. The numerator of the first fraction can be factored by taking out a factor of 9x^2, it will result in \(9x^2(5x - 1)\). The denominator of the second fraction can be simplified to 6x.

Step 2: Perform the multiplication

Multiplying the fractions gets \(\frac{9x^{2}(5x - 1)}{x} \cdot \frac{2}{6x}\).

Step 3: Simplify by cancelling the common factors

Then, you can cancel down : The common factors in the numerator and denominator are 2, \(x\), and \(3x\), so after cancelling these factors out, we get: \(\frac{3(5x-1)}{1}\). This simplification is valid only if \(x ≠ 0\) and \(x ≠ 5\), otherwise the original expressions would be undefined.

Step 4: Final simplification

Finally, the fraction simplifies to \(15x-3\)

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when simplifying algebraic expressions. It involves rewriting a polynomial as a product of its factors, which are often simpler expressions. This step can significantly reduce the complexity of a problem by pulling out common factors.

When dealing with the polynomials in this exercise, we started by factoring the expression \(45x^3 - 9x^2\) in the numerator of the first fraction. Both terms share \(9x^2\) as a common factor.

So, we pull out \(9x^2\), simplifying the expression to \(9x^2(5x - 1)\).

Understanding factoring is essential as it allows for easier manipulation of expressions and serves as a foundation for simplifying complex algebraic equations.

Fraction Multiplication

Multiplying fractions is a straightforward process, but it can become complicated without a strong understanding of each step. In algebra, fractions often feature polynomials, making it essential to handle each term correctly.

To multiply two fractions, simply multiply their numerators together and their denominators together. In this case, multiplying the simplified factors from our first fraction with those of the second yields the expression \(\frac{9x^{2}(5x - 1)}{x} \cdot \frac{2}{6x}\).

This multiplication sets the stage for simplification by making it easier to identify and cancel common factors in the resulting expression. Multiplying fractions always highlights the importance of precise arithmetic.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to understand and work with. Once you've multiplied fractions, the next step is to simplify, which often involves canceling out common factors.

For our given problem, after multiplying the fractions, we notice that both expressions share the factors of \(2\), \(x\), and \(3x\).

Cancelling these common factors results in \(\frac{3(5x-1)}{1}\), which simplifies further to \(15x - 3\).

Keep in mind, simplification is valid as long as the variables do not take on values that zero out the denominators in the original form, here meaning that \(x eq 0\) and \(x eq 5\).

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. They require careful manipulation and understanding to ensure correct simplification.

In the given exercise, we dealt with fractions that include variables and polynomials in both parts of the fraction.

This adds a layer of complexity, especially when determining common denominators and factoring. A key constraint with algebraic fractions is avoiding zero in the denominator, crucial for maintaining the validity of the expression.

Understanding how to simplify and multiply algebraic fractions empowers you to handle various algebraic challenges, opening the door to solving even more complicated expressions effectively.