Problem 42 - Elementary and Intermediate Algebra

Question

Expand each binomial. See Example 3. $$ (3 x-2)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \( 27x^3 - 54x^2 + 36x - 8 \).

Step 1: Apply the Binomial Theorem

The binomial theorem allows us to expand powers of binomials. It states that for any positive integer \( n \), \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \). In this case, let \( a = 3x \), \( b = -2 \), and \( n = 3 \).

Step 2: Calculate the Binomial Coefficients

Compute the binomial coefficients \( \binom{3}{0} \), \( \binom{3}{1} \), \( \binom{3}{2} \), and \( \binom{3}{3} \). These coefficients are given by the combinations formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Thus, we have: \( \binom{3}{0} = 1 \), \( \binom{3}{1} = 3 \), \( \binom{3}{2} = 3 \), \( \binom{3}{3} = 1 \).

Step 3: Expand Using the Binomial Terms

Now substitute \( a = 3x \) and \( b = -2 \) into the binomial terms for each coefficient. The expanded expression is: \[ \binom{3}{0}(3x)^3(-2)^0 + \binom{3}{1}(3x)^2(-2)^1 + \binom{3}{2}(3x)^1(-2)^2 + \binom{3}{3}(3x)^0(-2)^3 \].

Step 4: Simplify the Terms

Calculate each term individually: - First term: \( 1 \cdot (3x)^3 \cdot (-2)^0 = 27x^3 \) - Second term: \( 3 \cdot (3x)^2 \cdot (-2) = -54x^2 \) - Third term: \( 3 \cdot (3x) \cdot 4 = 36x \) - Fourth term: \( 1 \cdot 1 \cdot (-8) = -8 \). Combine these results to get the complete expansion.

Step 5: Write the Final Expanded Form

Combine the simplified terms to form the expanded polynomial: \[27x^3 - 54x^2 + 36x - 8\].

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial raised to a power as a polynomial. Polynomials are algebraic expressions made up of terms that include variables raised to a whole number exponent, coefficients, and constants. When we're expanding a binomial like \[(3x - 2)^3\], we are essentially breaking it down into a sum of terms.

The Binomial Theorem is a handy tool used for expansion. The theorem lets us express a binomial raised to the n-th power as a sum of terms involving binomial coefficients. We identify each term by a combination of the elements of the binomial, multiplied by these coefficients.

This theorem gives us a structured strategy to expand polynomials efficiently and accurately. It's extremely useful in algebra and helps simplify complex expressions easily.

Binomial Coefficients

Binomial coefficients are crucial components when expanding binomials. These coefficients are represented as \( \binom{n}{k} \) and are determined by specific formulas depending on the values of n and k. The combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) helps in identifying these coefficients.
In a binomial expansion, each term's coefficient can be found using the binomial coefficient. Let's expand \((3x-2)^3\):

• For \( k = 0 \), we have \( \binom{3}{0} = 1 \)
• For \( k = 1 \), we have \( \binom{3}{1} = 3 \)
• For \( k = 2 \), we have \( \binom{3}{2} = 3 \)
• For \( k = 3 \), we have \( \binom{3}{3} = 1 \)

These coefficients guide the expansion by dictating the multiplicative factor for each term.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the expression \((3x-2)^3\), we identify two elements: the variable term \(3x\) and the constant \(-2\).
When we expand the expression using the Binomial Theorem, each term in our expanded polynomial is essentially an algebraic expression. For instance, the first term \(27x^3\) is derived by computing \((3x)^3\), a power of an algebraic expression. Subsequent terms involve the strategic multiplication of binomial coefficients, powers of individual terms, and combinations of constants and variables.
Remember, the key to working effectively with algebraic expressions is understanding how to manipulate these variables and constants, which forms the basis of most algebraic operations. Managing these expressions correctly helps in solving equations and expanding binomials efficiently.