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

Verified step by step guidance

1. Recognize that the expression $(2x-3{)}^3$ is a binomial raised to a power, which can be expanded using the Binomial Theorem.
2. The Binomial Theorem states that $(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$. In this case, identify $a=2x$, $b=-3$, and $n=3$.
3. Apply the Binomial Theorem: $(2x-3{)}^3=\sum _{k=0}^3(\genfrac{}{}{0ex}{}{3}{k})(2x{)}^{3-k}(-3{)}^k$.
4. Calculate each term of the expansion: $(\genfrac{}{}{0ex}{}{3}{0})(2x{)}^3(-3{)}^0$, $(\genfrac{}{}{0ex}{}{3}{1})(2x{)}^2(-3{)}^1$, $(\genfrac{}{}{0ex}{}{3}{2})(2x{)}^1(-3{)}^2$, and $(\genfrac{}{}{0ex}{}{3}{3})(2x{)}^0(-3{)}^3$.
5. Simplify each term and combine them to get the expanded form of $(2x-3{)}^3$.

Key Concepts

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 allows for the calculation of each term in the expansion without direct multiplication.

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. When expanding a binomial like (2x - 3)^3, the result will be a cubic polynomial. Understanding the characteristics of cubic functions, such as their shape and behavior, is essential for interpreting the results of the expansion.

Algebraic Manipulation

Algebraic manipulation involves the process of rearranging and simplifying algebraic expressions using various mathematical operations. This includes applying the distributive property, combining like terms, and factoring. Mastery of these techniques is crucial for effectively expanding polynomials and simplifying the resulting expressions, as seen in the expansion of (2x - 3)^3.