Questions: Find the product. (m-p)^4 (m-p)^4= (Simplify your answer.)

Solution Steps

To find the product \((m-p)^4\), we need to expand the expression using the binomial theorem. The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = m\), \(b = -p\), and \(n = 4\).

1. Identify the values of \(a\), \(b\), and \(n\).
2. Use the binomial theorem to expand \((m - p)^4\).
3. Simplify the resulting expression.

We start with the expression \((m - p)^4\).

Using the binomial theorem, we expand \((m - p)^4\) as follows: \[ (m - p)^4 = \sum_{k=0}^{4} \binom{4}{k} m^{4-k} (-p)^k \]

Calculating each term in the expansion:

• For \(k = 0\): \(\binom{4}{0} m^4 (-p)^0 = m^4\)
• For \(k = 1\): \(\binom{4}{1} m^3 (-p)^1 = -4m^3p\)
• For \(k = 2\): \(\binom{4}{2} m^2 (-p)^2 = 6m^2p^2\)
• For \(k = 3\): \(\binom{4}{3} m^1 (-p)^3 = -4mp^3\)
• For \(k = 4\): \(\binom{4}{4} m^0 (-p)^4 = p^4\)

Combining all the terms gives us: \[ (m - p)^4 = m^4 - 4m^3p + 6m^2p^2 - 4mp^3 + p^4 \]

Final Answer