Problem 14 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 5 a 2 b-115 a 3(b-1) 2 $$

Step-by-Step Solution

Answer

The simplified form is \(5a^2b - 115a^3b^2 + 230a^3b - 115a^3\).

Step 1: Identify the Terms

In the expression \(5a^2b - 115a^3(b-1)^2\), we identify two terms: the first being \(5a^2b\) and the second being \(-115a^3(b-1)^2\). We aim to simplify these terms by expanding and combining like terms.

Step 2: Expand the Second Term

Expand the second term \(-115a^3(b-1)^2\) using the binomial formula. The term \((b-1)^2\) expands to \(b^2 - 2b + 1\). Thus, \(-115a^3(b-1)^2 = -115a^3(b^2 - 2b + 1)\).

Step 3: Distribute the Second Term

Distribute \(-115a^3\) across the expanded result: \[-115a^3 (b^2 - 2b + 1) = -115a^3b^2 + 230a^3b - 115a^3.\] Note the signs and coefficients for each resulting term.

Step 4: Write the Expression

Re-write the original expression with the expanded second term: \[5a^2b - 115a^3b^2 + 230a^3b - 115a^3.\] We now have a polynomial in terms of \(a\) and \(b\).

Step 5: Combine like Terms

Look for any like terms in the expression. In this case, there are no like terms among the coefficients and variable combinations, so further simplification in terms of combining like terms isn't possible.

Key Concepts

Binomial Expansion

To simplify polynomial expressions like the one in the exercise, understanding binomial expansion is crucial. When you come across a binomial term squared, such as \((b-1)^2\), you can expand it using the binomial formula: \((x-y)^2 = x^2 - 2xy + y^2\). In our example, set \(x = b\) and \(y = 1\), thus:

• \(b^2\) is the square of the first term.
• \(-2b\) represents twice the product of the two terms.
• \(+1\) is the square of the second term.

This gives us the expanded form \(b^2 - 2b + 1\). This expansion is often the first step in simplifying expressions that involve a binomial raised to a power. By expanding binomials, you turn them into a form that is easier to work with in further calculations.

Distributing Terms

Once you've expanded the binomial term, it's essential to distribute any coefficients or additional terms across it. In our case, the expression \(-115a^3(b^2 - 2b + 1)\) involves distributing \(-115a^3\) to each term inside the parentheses. Here's how it's done:

• Multiply \(-115a^3\) by each term in the expanded binomial: \(b^2\), \(-2b\), and \(+1\).
• For \(-115a^3 \times b^2 = -115a^3b^2\).
• For \(-115a^3 \times (-2b) = +230a^3b\) (remember, the negative signs cancel to become positive).
• Finally, \(-115a^3 \times 1 = -115a^3\).

Distributing terms is a fundamental operation in algebra that helps in simplifying, expanding, and reorganizing expressions.

Combining Like Terms

The last step in simplifying the polynomial involves combining like terms, which means simplifying by grouping terms that have the same variables raised to the same powers. In the given expression \(5a^2b - 115a^3b^2 + 230a^3b - 115a^3\), you need to check for such terms.
Like terms have the exact same variables with identical exponents. For instance, \(a^2b\) and \(a^2b\) are like terms, but \(a^2b\) and \(a^3b\) are not.
In our given expression, each term is unique regarding its variable and exponent configuration. Hence, there are no like terms to combine. The simplification process ends here in this particular scenario. However, this step is crucial in many expressions where combining like terms significantly reduces the complexity of the polynomial.