Problem 50 Find each product. $$ (3 m+1... [FREE SOLUTION]

Chapter 4: Problem 50

Find each product. $$ (3 m+1)^{3} $$

Short Answer

Expert verified

27m^3 + 27m^2 + 9m + 1

Step by step solution

Identify the formula

Recognize that this is a binomial raised to a power. Use the binomial theorem: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \]. Set $ a = 3m $, $ b = 1 $, and $ n = 3 $.

Apply the binomial theorem

Expand using $ n = 3 $, $ a = 3m $, and $ b = 1 $: \[ (3m+1)^3 = \binom{3}{0}(3m)^3(1)^0 + \binom{3}{1}(3m)^2(1)^1 + \binom{3}{2}(3m)^1(1)^2 + \binom{3}{3}(3m)^0(1)^3 \].

Evaluate each term

Compute each binomial coefficient and the corresponding powers:$ \binom{3}{0}(3m)^3(1)^0 = 1 \times 27m^3 \times 1 = 27m^3 $, $ \binom{3}{1}(3m)^2(1)^1 = 3 \times 9m^2 \times 1 = 27m^2 $, $ \binom{3}{2}(3m)^1(1)^2 = 3 \times 3m \times 1 = 9m $, and $ \binom{3}{3}(3m)^0(1)^3 = 1 \times 1 \times 1 = 1 $.

Sum the expanded terms

Add all the evaluated terms together: \[ 27m^3 + 27m^2 + 9m + 1 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expansion

Polynomial expansion is a key concept in algebra. It involves expressing a polynomial raised to a power as a sum of terms. The binomial theorem provides a systematic way to do this for binomial expressions. For instance, expanding $(3m+1)^{3}$ yields additional terms that can be calculated using specific formulas. Understanding polynomial expansion requires familiarity with terms, coefficients, and power rules.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply each term in the binomial expansion. They are represented as $\binom{n}{k}$, where $n$ is the power, and $k$ is the term's position. These coefficients can be found using the formula:

\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

In our exercise, the coefficients are $\binom{3}{0}, \binom{3}{1}, \binom{3}{2},$ and $\binom{3}{3}$, which evaluate to 1, 3, 3, and 1, respectively. These coefficients help determine the weight of each term in the expanded form.

Algebraic Expressions

Understanding how to manipulate algebraic expressions is critical when expanding binomials. An algebraic expression includes variables, constants, and mathematical operations. In our problem, $3m$ and 1 are the parts of the binomial $(3m+1)$. During expansion, we raise these parts to specific powers and multiply them by the binomial coefficients. For example:

$(3m)^3 = 27m^3 $
$(3m)^2 = 9m^2 $
$(3m)^1 = 3m $
$(3m)^0 = 1 $

Combining these with the binomial coefficients results in terms like $27m^3, 27m^2, 9m,$ and 1, which sum to form the final expanded polynomial: $27m^3 + 27m^2 + 9m + 1$.

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91影视

Find each product. $$ (3 m+1)^{3} $$

Short Answer

Step by step solution

Identify the formula

Apply the binomial theorem

Evaluate each term

Sum the expanded terms

Key Concepts

Polynomial Expansion

Binomial Coefficients

Algebraic Expressions

One App. One Place for Learning.

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