Problem 47 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ (b+1)^{4} \cdot \frac{(b-7)^{3}}{b+1} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$ (b+1)^{4} \cdot \frac{(b-7)^{3}}{b+1} $$ Answer: The simplified expression is: $$ (b+1)^{3} \cdot (b-7)^{3} $$

Step 1: Write out the given expression

We are given the following expression to simplify: $$ (b+1)^{4} \cdot \frac{(b-7)^{3}}{b+1} $$

Step 2: Cancel common factors

We notice that \((b+1)\) is both in the numerator and the denominator, so we can cancel it: $$ ((b+1)^{4} \cdot (b-7)^{3}) \div (b+1) \implies (b+1)^{3} \cdot (b-7)^{3} $$ We used the rule \((a^m)^n = a^{mn}\) to simplify the expression.

Step 3: Expand the expression (optional)

The expression is now simplified, but we can expand it if needed. However, expanding it would make it more complicated and hard to read. If the question does not require expansion, we can leave the expression as is: $$ (b+1)^{3} \cdot (b-7)^{3} $$ This is the simplified expression for the given problem.

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division with numbers. In the context of algebra, it's utilized to simplify expressions where one polynomial is divided by another. This concept is essential when you're trying to reduce the complexity of an algebraic fraction or when factoring polynomials.

One of the basic applications of polynomial division is cancelling common factors in the numerator and denominator, as seen in the textbook exercise, where \(b+1\) is present in both and therefore, can be simplified. By recognizing this, the expression \( (b+1)^4 \cdot (b-7)^3 \) divided by \(b+1\) was reduced to \( (b+1)^3 \cdot (b-7)^3 \), which simplified the overall problem.

Exponent Rules

Understanding exponent rules is crucial for simplifying algebraic expressions that involve powers. When you have the same base being multiplied with different exponents, you add the exponents. Conversely, when you divide, you subtract the exponents. Other important exponent rules include the power of a product rule and the power of power rule.

In the example, we encounter the power of power rule. Initially, the problem involves \( (b+1)^4 \cdot (b-7)^3 \) divided by \(b+1\). The rule \(a^{m} \cdot a^{n} = a^{m+n}\) does not apply here because the bases differ. However, when dividing powers with the same base, such as \(a^m \div a^n\), we use the rule \(a^m \div a^n = a^{m-n}\), which correctly simplifies \( (b+1)^4 \div (b+1)\) to \( (b+1)^3\).

Binomial Expansion

Binomial expansion involves expanding expressions that include binomials raised to a power, such as \( (a+b)^n\). This can be done using the Binomial Theorem, which provides a formula for the expansion or Pascal's triangle. The resulting expression consists of multiple terms, each a product of binomial coefficients and the variables a and b raised to various powers.

While binomial expansion can be utilized to expand \( (b+1)^3 \cdot (b-7)^3 \), as in our exercise, it can make the expression more cumbersome. In practice, we often leave the expression in its factored form for simplicity, unless specifically required to expand. Additionally, expanding without necessity can introduce errors and complicate further algebraic manipulation.