Problem 40 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (3-2 x)^{5} $$

Step-by-Step Solution

Answer

The expanded expression is \(243 - 810x + 1080x^2 - 720x^3 + 240x^4 - 32x^5\).

Step 1: Understand Pascal's Triangle

Pascal's Triangle is a triangular array of binomial coefficients. For the expansion of \((a + b)^n\), the \( n^{th} \) row of Pascal's Triangle provides the coefficients. We need the coefficients for \((3 - 2x)^5\), which needs the 5th row of Pascal's Triangle: 1, 5, 10, 10, 5, 1.

Step 2: Apply Binomial Theorem

The Binomial Theorem states \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, \(a = 3\), \(b = -2x\), and \(n = 5\). Substitute these into the theorem: \((3 - 2x)^5 = \sum_{k=0}^{5} \binom{5}{k} 3^{5-k} (-2x)^k\).

Step 3: Calculate Each Term

Using the coefficients from Pascal's Triangle and the binomial theorem, calculate each term:- First term \( = \binom{5}{0} 3^5 (-2x)^0 = 1 \times 243 \times 1 = 243\).- Second term \( = \binom{5}{1} 3^4 (-2x)^1 = 5 \times 81 \times (-2x) = -810x\).- Third term \( = \binom{5}{2} 3^3 (-2x)^2 = 10 \times 27 \times 4x^2 = 1080x^2\).- Fourth term \( = \binom{5}{3} 3^2 (-2x)^3 = 10 \times 9 \times (-8x^3) = -720x^3\).- Fifth term \( = \binom{5}{4} 3^1 (-2x)^4 = 5 \times 3 \times 16x^4 = 240x^4\).- Sixth term \( = \binom{5}{5} 3^0 (-2x)^5 = 1 \times 1 \times (-32x^5) = -32x^5\).

Step 4: Combine and Write the Expanded Expression

Add all the calculated terms together to find the expansion: \((3 - 2x)^5 = 243 - 810x + 1080x^2 - 720x^3 + 240x^4 - 32x^5\).

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential tool in algebra, used for the expansion of powers of a binomial expression. A **binomial** is simply a sum of two terms, like \((a + b)\). If you need to expand \((a + b)^n\), where "n" is a non-negative integer, the theorem makes this task systematic.
The theorem states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] Here, \(\binom{n}{k}\) is a **binomial coefficient** which gives the number of ways to choose "k" successes in "n" trials, and the symbols \(a^{n-k}\) and \(b^k\) represent the decreasing and increasing powers of "a" and "b", respectively.
This method of expansion saves you the hassle of multiplying the term by itself "n" times, providing a quicker and more systematic approach.

Binomial Coefficients

**Binomial coefficients** are the key numbers found in Pascal's Triangle and represent the coefficients in the expansion given by the Binomial Theorem. These coefficients are denoted by \(\binom{n}{k}\), known also as "n choose k," and have a simple formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] Here, "!" refers to factorial, which means multiplying a series of descending natural numbers. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).
Pascal's Triangle is a valuable tool for quickly finding these coefficients without calculation. To form it, start with a single "1" at the top, and then build down in a triangular fashion with each interior number being the sum of the two directly above it.

• The top of the triangle is row 0, which corresponds to \((a + b)^0\), yielding only 1.
• The next row, 1, which corresponds to \((a + b)^1\), has the numbers 1 and 1.
• The pattern continues, such that the 5th row, used for \((3 - 2x)^5\), is 1, 5, 10, 10, 5, 1.

Polynomial Expansion

**Polynomial Expansion** refers to the process of expanding a binomial raised to a power into a sum of terms. This results in a polynomial, which is an expression consisting of variables raised to various powers and multiplied by coefficients. In the context of the exercise \((3 - 2x)^5\), we use the binomial theorem to expand this binomial into a polynomial.
Each term in the expansion comes from substituting the variables into the binomial theorem:

• Start with finding the binomial coefficients for the specific expansion, using either Pascal's Triangle or the binomial coefficient formula.
• Substitute the coefficients and variables into \(\binom{n}{k} a^{n-k} b^k\).
• Simplify each term by calculating the powers and products involved.

The polynomial expansion of \((3 - 2x)^5\) results in six terms: 243, -810x, 1080x², -720x³, 240x⁴, and -32x⁵.
Once all terms are calculated, combine them to form the expanded polynomial expression, which in this case is \(243 - 810x + 1080x^2 - 720x^3 + 240x^4 - 32x^5\).