Problem 30 - Beginning and Intermediate Algebra

Question

Before expanding \((t-4)^{6}\) using the binomial theorem, how should the binomial be rewritten?

Step-by-Step Solution

Answer

The given binomial, \((t-4)^6\), should be rewritten in the form of \((x+y)^n\) before expanding using the binomial theorem. To do this, rewrite it as \((t + (-4))^6\), where x = t, y = -4, and n = 6.

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any natural number n and real number x, y, we have: \[(x + y)^n = \sum_{k=0}^n{n \choose k}x^{n-k}y^k\] where \({n \choose k}\) is the binomial coefficient and can be calculated as: \({n \choose k} = \frac{n!}{k!(n-k)!}\), where "n!" denotes n factorial (the product of all positive integers up to n).

Step 2: Rewrite the given binomial to fit the Binomial Theorem

In order to rewrite the given binomial, \((t-4)^6\), to fit the Binomial Theorem, we need to express it in the form \((x+y)^n\). We can do this by noting that the expression can be equivalently written as: \[(t + (-4))^6\]

Step 3: Identify x, y, and n in the rewritten binomial

Now that we rewrote the binomial in the form of \((x+y)^n\), we can identify x, y, and n as follows: - x = t - y = -4 - n = 6 Now the binomial has been rewritten in a form suitable for applying the binomial theorem: \((t + (-4))^6\).