Problem 66 - Elementary Algebra

Question

For the following problems, note how many: $$ 3(a+8) \text { 's in } 6 x^{2}(a+8)^{3}(a-8) ? $$

Step-by-Step Solution

Answer

Answer: There are \(0\) instances of \(3(a+8)\) in the given expression.

Step 1: Expand the expression

Expand the expression \(6x^2(a+8)^3(a-8)\) by applying the exponent and distributing: $$ 6x^2(a+8)^3(a-8) = 6x^2((a+8)(a+8)(a+8))(a-8) $$

Step 2: Simplify the expression

Now, simplify the expression by carrying out the multiplication: $$ 6x^2\left((a^3+24a^2+192a+1024)(a-8)\right) $$

Step 3: Distribute (a-8) to the expanded expression

Next, distribute \((a-8)\) to the expanded expression: $$ 6x^2(a^4-8a^3+24a^3-192a^2+192a-1536a+1024a-8192) $$

Step 4: Combine like terms

Now, combine the like terms to further simplify the expression: $$ 6x^2(a^4+16a^3-192a^2+1280a-8192) $$

Step 5: Identify the factor 3(a+8)

Lastly, we need to identify the factor \(3(a+8)\) in the simplified expression. Notice that this factor is not present in the simplified expression; therefore, there are \(0\) instances of \(3(a+8)\) in the expression \(6x^2(a+8)^3(a-8)\). So, the final answer is that there are \(0\) instances of \(3(a+8)\) in the given expression.

Key Concepts

Expansion and Distribution

Expansion and distribution are fundamental concepts when dealing with algebraic expressions, especially when you have terms grouped together inside parentheses. In this case, we have the expression

• \((a+8)^3\)
• \((a-8)\)

These are being multiplied together within the larger expression.
To simplify this, you start by expanding the cube, \((a+8)^3 = (a+8)(a+8)(a+8)\).
This means you multiply the expression by itself three times.
Next, you'll need to distribute each term inside the parentheses.
This involves applying the distributive property, which allows us to remove the brackets by multiplying each term inside the parentheses by the other term. For example, multiplying \((a+8)\) by \((a+8)\) results in \(a^2 + 16a + 64\).

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions with more than one term.
When multiplying polynomials, every term in one polynomial must be multiplied by every term in the other polynomial.
In this exercise, we are multiplying \((a+8)^3\) by \((a-8)\).

• Start with one part of the expression: expand \((a+8)^3\) to obtain \((a^3 + 24a^2 + 192a + 1024)\).
• Next, use this expansion to multiply by \((a-8)\).
• Multiply each term in the expanded \((a+8)^3\) by every term in \((a-8)\).
• This results in polynomials in a single expression which you'll need to simplify further.

This process requires attention to distributing the terms carefully and organizing them into a clear order. Once all terms are multiplied, you'll have several terms that you'll need to combine.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, ensuring there are no like terms left to combine. After expanding and multiplying in the previous steps, you have resulting terms such as:

• \(a^4\)
• \(16a^3\)
• \(-192a^2\)
• \(1280a\)
• \(-8192\)

Combining like terms involves identifying terms with the same variable raised to the same power and summing their coefficients.
For instance, if you have \(24a^3 - 8a^3\), you combine them to get \(16a^3\).
This makes the expression cleaner and easier to work with. Note that an important part of simplification is to check for common factors like \(3(a+8)\).
In this solution, we see that \(3(a+8)\) does not appear, which helped conclude that there were no instances of this factor in the expression.
This wraps up the simplification process, leading to a clear and concise final expression.