Problem 69 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ b^{5} x^{2}(2 b x-11) $$

Step-by-Step Solution

Answer

Question: Multiply the given expression \(b^{5} x^{2}(2 b x-11)\) and simplify. Answer: \(2b^{6}x^{3} - 11b^{5}x^{2}\)

Step 1: Apply distributive property

To start, we'll multiply \(b^{5} x^{2}\) by each term within the parentheses: $$ b^{5} x^{2} (2bx - 11) = (b^{5} x^{2} \cdot 2bx) - (b^{5} x^{2} \cdot 11) $$

Step 2: Multiply the terms

Now, we'll apply the multiplication to individual terms: $$ (b^{5} x^{2} \cdot 2bx) - (b^{5} x^{2} \cdot 11) = 2b^{6}x^{3} - 11b^{5}x^{2} $$ Notice that, since we have to multiply terms with the same base, we add their exponents.

Step 3: Final answer

The given expression \(b^{5} x^{2}(2 b x-11)\) can be simplified to: $$ 2b^{6}x^{3} - 11b^{5}x^{2} $$ The final result is a simplified expression with no like terms to combine.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental rule in algebra that is especially useful when dealing with expressions inside parentheses. It allows us to multiply each term within the parentheses by a factor outside them.
For example, when you have an expression like \( a(b + c) \), the distributive property tells us to multiply \( a \) by both \( b \) and \( c \):

• First we do \( a \times b \)
• Then we do \( a \times c \)

This results in \( ab + ac \).
Using this property helps to simplify complex expressions and is an important step in solving equations. When working with polynomial expressions, it's crucial to apply the distributive property accurately to ensure correct results.

Multiplication of Polynomials Explained

Multiplying polynomials involves applying the distributive property but on a larger, more complex scale. Each term in the first polynomial needs to be multiplied by each term in the second polynomial.
This process seems cumbersome at first, but it’s just breaking down the problem into simpler, bite-sized steps. In our original problem, this process involved:

• First multiplying \( b^{5}x^{2} \) by \( 2bx \), which gave \( 2b^{6}x^{3} \)
• Next multiplying \( b^{5}x^{2} \) by \( 11 \), which results in \( 11b^{5}x^{2} \)

The key point is to multiply coefficients together and add the exponents for terms with the same base. This step-by-step method ensures accuracy and helps to keep track of the terms as they grow more complex.

Simplifying Expressions

Simplifying expressions is all about combining like terms and reducing the expression to its simplest form. After applying the distributive property and multiplication of terms in an expression, like terms, if present, need to be combined.
In the example we have, the final expression becomes \( 2b^{6}x^{3} - 11b^{5}x^{2} \).
In this expression, there are no like terms to combine since the exponents on the bases differ, and thus it’s already in its simplest form. Simplifying involves:

• Identifying like terms, which are terms that have the same variables raised to the same power.
• Adding or subtracting the coefficients of these terms.

Simplifying helps to present expressions in a clearer, more concise form, and makes it easier to perform further calculations if needed.