(2a-5+b)5

Jasonbradley

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Jun 16, 2024

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Expanding the Expression (2a - 5 + b)5

This expression represents a polynomial raised to the power of 5. To fully expand it, we need to apply the binomial theorem or use repeated multiplication.

Understanding the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n. It states:

(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k

Where:

• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
• n is the power to which the binomial is raised
• k is an index that ranges from 0 to n

Applying the Binomial Theorem

To apply the binomial theorem to our expression (2a - 5 + b)5, we need to consider it as a sum of three terms:

• (2a)
• (-5)
• (b)

The theorem will then involve combinations of these three terms, raised to various powers.

Expanding the Expression

Expanding (2a - 5 + b)5 using the binomial theorem is quite complex, resulting in a long expression with many terms. Here's a simplified illustration of the process:

1. Identify the terms: (2a), (-5), and (b).
2. Apply the binomial theorem for each term: This will involve calculating binomial coefficients and raising each term to different powers.
3. Combine the resulting terms: The final expansion will have a sum of all the terms obtained from each term in the original expression.

Simplification and Result

The final expanded form will be a polynomial with numerous terms. You can simplify this expression by combining like terms and arranging them in descending order of their powers.

Alternative Approach: Repeated Multiplication

Instead of using the binomial theorem, we can expand the expression by repeatedly multiplying it out. This can be a more straightforward approach but becomes increasingly complex as the power increases.

For example:

• (2a - 5 + b)5 = (2a - 5 + b)(2a - 5 + b)(2a - 5 + b)(2a - 5 + b)(2a - 5 + b)

This method requires multiplying the first two brackets, then multiplying the result by the next bracket, and so on. The final result will be the same as the one obtained using the binomial theorem.

Important Note:

Expanding (2a - 5 + b)5 using either method will result in a long and complex expression. The specific details of the expanded form will depend on the careful application of the binomial theorem or repeated multiplication.