Answered

The terms of the binomial expansion are written using the patterns.

$(m+2)^4=(1)\left(m^4\right)\left(2^0\right)+(4)\left(m^3\right)\left(2^1\right)+(6)\left(m^2\right)\left(2^2\right)+(4)\left(m^1\right)\left(2^3\right)+(1)\left(m^0\right)\left(2^4\right)$



Simplify each term to complete the expansion.

$(m+2)^4=m^4+\square m^3+\square m^2+\square m+\square$

Verified Answer :

  • Simplify the terms of the binomial expansion \((m+2)^4\) individually.
  • The first term is \(1 \cdot m^4 \cdot 1 = m^4\).
  • The second term is \(4 \cdot m^3 \cdot 2 = 8m^3\).
  • The third term is \(6 \cdot m^2 \cdot 4 = 24m^2\).
  • The fourth term is \(4 \cdot m^1 \cdot 8 = 32m\).
  • The fifth term is \(1 \cdot 1 \cdot 16 = 16\).
  • The final expansion is \((m+2)^4 = m^4 + 8m^3 + 24m^2 + 32m + 16\).
  • Therefore, the polynomial is \(m^4 + \boxed{8}m^3 + \boxed{24}m^2 + \boxed{32}m + \boxed{16}\).

Explanation

  1. Analyzing the Binomial Expansion Pattern

The problem asks us to simplify the binomial expansion of \((m+2)^4\), given by the expression:
\[(m+2)^4=(1)(m^4)(2^0)+(4)(m^3)(2^1)+(6)(m^2)(2^2)+(4)(m^1)(2^3)+(1)(m^0)(2^4)\]
To express this in the form \(m^4 + Cm^3 + Dm^2 + Em + F\), we need to simplify each individual term by performing the indicated multiplication. We know that \(m^0 = 1\) and any number raised to the power of 0 is 1. Thus, \(2^0 = 1\) and \(m^0 = 1\).

  1. Simplifying Each Term

We simplify each of the five terms one by one:

  • The first term is \((1)(m^4)(2^0) = 1 \cdot m^4 \cdot 1 = m^4\).
  • The second term is \((4)(m^3)(2^1) = 4 \cdot m^3 \cdot 2 = 8m^3\).
  • The third term is \((6)(m^2)(2^2) = 6 \cdot m^2 \cdot 4 = 24m^2\).
  • The fourth term is \((4)(m^1)(2^3) = 4 \cdot m \cdot 8 = 32m\).
  • The fifth term is \((1)(m^0)(2^4) = 1 \cdot 1 \cdot 16 = 16\).
  1. Forming the Final Polynomial

Now, we combine the simplified terms to form the final polynomial:
\[(m+2)^4 = m^4 + 8m^3 + 24m^2 + 32m + 16\]
Comparing this to the target form \(m^4 + \boxed{\phantom{0}}m^3 + \boxed{\phantom{0}}m^2 + \boxed{\phantom{0}}m + \boxed{\phantom{0}}\), we identify the missing values as 8, 24, 32, and 16.

Examples

The binomial expansion is a fundamental tool in algebra and probability. For instance, if you are designing a game where a character has a 50% chance to succeed or fail (like tossing a coin), the outcomes of 4 trials are described by the expansion of \((0.5+0.5)^4\). By understanding how to expand \((m+2)^4\), you gain the ability to calculate probabilities, model growth processes, or simplify complex algebraic expressions efficiently in fields like computer science, finance, and physics.

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