In Exercises 15–58, find each product. (x+2)3

Ch. P - Fundamental Concepts of Algebra

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Ch. P - Fundamental Concepts of Algebra

Problema 52

Capitolo 1, Problema 52

In Exercises 15–58, find each product. (x+2)³

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Recognize that the expression $(x+2)^3$ represents the cube of a binomial. To expand this, we use the Binomial Theorem, which states $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$.

Identify the components of the binomial: $a = x$, $b = 2$, and $n = 3$.

Apply the Binomial Theorem to expand $(x+2)^3$: $(x+2)^3 = \binom{3}{0}x^3(2^0) + \binom{3}{1}x^2(2^1) + \binom{3}{2}x^1(2^2) + \binom{3}{3}x^0(2^3)$.

Simplify each term using the binomial coefficients $\binom{n}{k}$, which are calculated as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. For example, $\binom{3}{0} = 1$, $\binom{3}{1} = 3$, $\binom{3}{2} = 3$, and $\binom{3}{3} = 1$.

Combine the simplified terms to write the expanded form of $(x+2)^3$. The result will be a polynomial with four terms.

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Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion can be systematically achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem provides a formula for calculating the coefficients of the expanded terms.

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Special Products - Cube Formulas

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d. In the context of the question, (x + 2)^3 represents a cubic function where the variable x is transformed by adding 2 before cubing. Understanding cubic functions is essential for recognizing their properties, such as their shape and the behavior of their graphs.

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Function Composition

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials to produce a new polynomial. This process requires distributing each term in the first polynomial to every term in the second polynomial, combining like terms afterward. In the case of (x + 2)^3, this means multiplying (x + 2) by itself three times, which illustrates the principles of both distribution and combining like terms.

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