Problem 32 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{-14 b^{2} c^{2}+21 b^{3} c^{3}-28 c^{3}}{-7 a^{2} c^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{-14 b^{2} c^{2}+21 b^{3} c^{3}-28 c^{3}}{-7 a^{2} c^{3}}$. Answer: $\frac{-2 b^{2} c^{2} + 3 b^{3} c^{3} - 4 c^{3}}{a^2 c^3}$

Step 1: Identify the common factors in the expression

Firstly, let's take a closer look at the numerator and denominator of the expression: $$ \frac{-14 b^{2} c^{2}+21 b^{3} c^{3}-28 c^{3}}{-7 a^{2} c^{3}} $$ We observe that there is a common factor of 7 in the numerator terms. Similarly, the denominator has a factor of 7.

Step 2: Factor out the common factors in the expression

Now, we factor out the common factor of 7 in the numerator terms and -7 in the denominator term (we choose to factor out -7 to easily eliminate the negative signs). $$ \frac{7(-2 b^{2} c^{2} + 3 b^{3} c^{3} - 4 c^{3})}{-7 (a^2 c^3)} $$

Step 3: Cancel out common factors

We can now cancel out the common factor of 7 from the numerator and the denominator. $$ \frac{-2 b^{2} c^{2} + 3 b^{3} c^{3} - 4 c^{3}}{a^2 c^3} $$

Step 4: Write the final simplified expression

Now we have simplified the expression completely, and we can write the final answer. $$ \frac{-2 b^{2} c^{2} + 3 b^{3} c^{3} - 4 c^{3}}{a^2 c^3} $$

Key Concepts

Simplifying Algebraic Expressions

When dealing with algebraic expressions, simplicity is our goal. Simplifying can make an expression easier to understand and work with. First, look for like terms—terms that have the same variable raised to the same power. These can often be combined by adding or subtracting coefficients. In the exercise provided, the numerator consists of terms that all contain a variable 'c' raised to different powers, but they do not qualify as like terms since the powers are not the same.

It's also vital to search for common factors across all terms, both in the numerator and denominator, as they can be factored out. In our example, '7' was a common factor and could be factored out. Remember, the greatest common factor can simplify expressions significantly, leading to an easier-to-manage result.

Factoring Polynomials

Factoring polynomials is like breaking down a complex structure into its building blocks—the prime factors. This process is crucial for simplifying expressions and solving equations. The idea is to express the polynomial as a product of its simplest polynomial parts. This can be achieved by identifying the greatest common factor (GCF) or using methods like grouping, the difference of squares, or the sum/product of roots for quadratics.

For instance, the exercise required factoring out the common factor of 7 from each term. Recognizing these factors can streamline complex algebraic operations and can sometimes reveal factors that cancel out, leading to further simplification.

Canceling Common Factors

Once you factor out common factors, you're often left with terms that appear in both the numerator and denominator of a fraction. As per the fundamental principles of algebra, any factor in the numerator and denominator that are identical can be canceled out. This is because anything divided by itself equals one. However, you must ensure that you only cancel factors, not terms, and that the factor is present across the entire numerator or denominator.

In the given exercise, after factoring out the common factor of 7, it was possible to cancel it out from both sides of the fraction, thus reducing the complexity of the expression. This step is a crucial simplification technique in algebra that helps in reducing fractions to their simplest form.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or denominator contain algebraic expressions. Simplifying them involves the same concepts as simplifying numerical fractions—factoring, reducing, and sometimes even expanding the expressions to find common denominators. The goal is always to make the expression as straightforward as possible.

Through the process seen in our exercise, we simplified a complex algebraic fraction by factoring to reveal common factors and canceling them out. When working with algebraic fractions, it's essential to be methodical and patient, simplifying step by step to avoid mistakes and maintain accuracy in the final result.