Problem 66 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ (b+5)^{3} \cdot \frac{(b+1)^{2}}{(b+5)^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{(b+5)^{3}(b+1)^{2}}{(b+5)^{2}}$$ and identify any restrictions on the variable b. Answer: The simplified expression is $$(b+5)(b+1)^{2}$$ with the restriction that $$b\neq -5$$.

Step 1: Identify the common terms

First, let's identify the common terms between the numerator and the denominator. In this case, we have \((b+5)^2\) present in both the numerator and the denominator. So, we could cancel out this term from the final expression.

Step 2: Cancel out the common terms

Now, we can cancel out the common terms \((b+5)^2\) in both the numerator and the denominator, providing we account for restrictions on b. The expression becomes: $$ \bigg(\frac{(b+5)^{3}}{(b+5)^{2}}\bigg) \cdot (b+1)^{2} $$

Step 3: Simplify the expression

Simplifying the expression after cancelling out the common terms, we get: $$ (b+5) \cdot (b+1)^{2} $$

Step 4: Identify restrictions on the variable

Since we cancelled out \((b+5)^2\) from the denominator, there is a restriction on values of b, for which the denominator becomes zero. In this case, the restriction is $$ b + 5 \neq 0 $$ which means $$ b \neq -5 $$ So, the simplified expression is valid for all values of b except b = -5.

Key Concepts

Polynomial Operations

Understanding polynomial operations is pivotal for simplifying algebraic expressions, which often include addition, subtraction, multiplication, and sometimes division of polynomials. In our example where we have \( (b+5)^{3} \cdot \frac{(b+1)^{2}}{(b+5)^{2}} \) as the expression to simplify, we are required to perform polynomial multiplication and division.

When performing operations on polynomials, it is important to recognize common terms or factors as they can often be simplified. Here, \( (b+5)^{2} \) is a common term in both the numerator and the denominator, which allows us to simplify the expression as a primary step. This significantly decreases the complexity of the problem. Remember that any term with an exponent indicates repeated multiplication of the term; for instance, \( (b+5)^{3} \) means \( (b+5) \cdot (b+5) \cdot (b+5) \) and can be simplified by cancelling common factors present in both the numerator and the denominator, provided they are not equal to zero (which would make the expression undefined).

Factoring Polynomials

The method of factoring polynomials is used to break down complicated polynomial expressions into products of simpler ones, which often reveal common factors. In factoring, we look for patterns, such as the difference of squares, perfect square trinomials, or common terms, and utilize techniques like grouping to factor the expression. In our original problem, the process of factoring isn't used, but if we had to factor the final simplified expression \( (b+5) \cdot (b+1)^{2} \), we would look for such patterns or common factors.

Correctly factoring polynomials is integral to simplifying complex algebraic fractions by revealing terms that can be cancelled. This makes understanding both the process of identifying common factors and utilizing different factoring techniques essential for simplifying polynomial expressions.

Algebraic Fractions

An algebraic fraction is a fraction where the numerator and/or the denominator are algebraic expressions. Simplifying them often involves factoring, reducing expressions to lowest terms, and cancelling out common factors. In the context of our example, \((b+5)^{3} \cdot \frac{(b+1)^{2}}{(b+5)^{2}}\), the numerator and the denominator both contain polynomials, and we simplify by reducing the common polynomial term, \( (b+5)^{2} \).

When simplifying algebraic fractions, always watch for restrictions that may arise due to the possibility of division by zero. In our problem, once we cancel out \( (b+5)^{2} \), we must remember that the cancellation is valid as long as the term cancelled is not equal to zero. Here, we conclude that \( b eq -5 \) to avoid a zero denominator, which is an important part of working with algebraic fractions and must be explicitly stated as part of the solution.