Question

Multiply: \((3 x-2)^{3}\)

Short answer

(3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8

Step-by-step solution

- Understand the Formula

To expand (a-b)^3, use the binomial theorem: (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Identify: a = 3x, and b = 2.

- Calculate Each Term

Substitute a = 3x and b = 2 into the formula: (3x - 2)^3 = (3x)^3 - 3(3x)^2(2) + 3(3x)(2^2) - 2^3.

- Simplify Each Term

(3x)^3 = 27x^3, 3(3x)^2(2) = 54x^2, 3(3x)(2^2) = 36x, 2^3 = 8. Therefore, (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8.

- Combine the Terms

Combine all terms to reach the final answer: (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8.

Key concepts

binomial theorem

The binomial theorem is a powerful tool in algebra, used to expand expressions that are raised to a power. The theorem states that any binomial raised to a positive integer exponent can be expanded into a sum involving terms of the form \(a^n b^m\). For example, to expand \( (3x - 2)^3 \), we use the formula \[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \].
This formula can save you a lot of time by providing a structured way to expand the binomial without tedious multiplication. By identifying \(a = 3x\) and \(b = 2\) in our problem, each term in the expansion can be handled easily.

• First term: \( (3x)^3 \)
• Second term: \(-3(3x)^2(2) \)
• Third term: \(+3(3x)(2^2) \)
• Fourth term: \(-2^3 \)

Plugging in these values, we can expand and simplify to find the final answer.

polynomial multiplication

Polynomial multiplication is another essential concept that helps in understanding binomial expansion better. When multiplying polynomials, you need to distribute each term in the first polynomial by each term in the second polynomial.
However, with binomial expansion, the binomial theorem makes this process much quicker. In our exercise, using polynomial multiplication to expand \( (3x - 2)^3 \) would involve multiplying \( (3x - 2) \) three times:
\[ (3x - 2)(3x - 2)(3x - 2) \] Without the binomial theorem, you would need to use the distributive property repeatedly.
The binomial theorem skips these steps by giving an immediate solution.
Remember, polynomial multiplication breaks down to consistently applying the distributive property to every term involved.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and subtraction). They form the building blocks of nearly all algebra problems, including our binomial expansion exercise.
Understanding how to manipulate and simplify these expressions is key. For example, in the expression \( (3x - 2)^3 \), we see several algebraic elements: \(3x\) and \(2\), combined using subtraction.
By using the binomial theorem, we are manipulating these elements to simplify the expression into a more manageable form.
The process of substitution (where \(a = 3x\) and \(b = 2\)) shows how variables and constants can be managed within larger expressions.
This concept is not just limited to binomial expansion; it is essential for solving a variety of algebraic problems.

exponents

Exponents, or powers, are a mathematical operation involving two numbers: the base and the exponent. They represent repeated multiplication. For instance, \( (3x)^3 = 27x^3 \).
When dealing with exponents in binomial expansion, you often have to simplify terms like \( (3x)^2 \), which means \( (3x)(3x) = 9x^2 \).
In our solution, exponents are used to expand each term accurately. Here's how exponents influence our problem:

• The term \( (3x)3 = 27x^3 \)
• The term \(-3(3x)^2(2) = -54x^2 \)
• The term \(+3(3x)(2^2) = 36x \)
• The term \(-2^3 = -8 \)

Understanding exponents is crucial for expanding and simplifying expressions like \( (3x - 2)^3 \). Remember, practice with different exponent rules helps in mastering these concepts.