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Chapter 9: Problem 41

Expand the binomial by using Pascal's Triangle to determine the coefficients. $$(2 t-s)^{5}$$

Short Answer

Expert verified
The expansion of the binomial \((2t - s)^5\) is: \(32t^5 + 160t^4s + 320t^3s^2 + 320t^2s^3 + 160ts^4 + 32s^5\).

Step by step solution

01

Construct Pascal's Triangle

For a binomial of the power 5, build Pascal鈥檚 Triangle up to the 5th row: \[1\ 1\ 1\ 1\ 2\ 1\ 1\ 3\ 3\ 1\ 1\ 4\ 6\ 4\ 1\ 1\ 5\ 10\ 10\ 5\ 1\] The numbers found in the 5th row of the triangle will be the coefficients of the expanded terms.
02

Assign values to the binomial expansion

Substitute the values for the variables 't' and 's' in the binomial expansion formula, \((a+b)^n = a^n + {n \choose 1}a^{n-1}b + {n \choose 2}a^{n-2}b^2 + ... + b^n\) where a = 2t, b = -s and n = 5.
03

Expand the binomial

On expanding the binomial, we get: \((2t - s)^5 = (2t)^5 - {5 \choose 1}(2t)^4(-s) + {5 \choose 2}(2t)^3(-s)^2 - {5 \choose 3}(2t)^2(-s)^3 + {5 \choose 4}(2t)(-s)^4 - (-s)^5\).
04

Simplify the terms

Simplify each term by carrying out the calculations: \(32t^5 + 160t^4s + 320t^3s^2 + 320t^2s^3 + 160ts^4 + 32s^5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pascal's Triangle
Pascal's Triangle is a simple yet powerful tool in algebra that helps to find coefficients in a binomial expansion. This triangular array of numbers is constructed by starting with a "1" at the top. The edges of the triangle are all "1"s. Each number inside the triangle is the sum of the two numbers directly above it. For example, to find the numbers of the 5th row, you look just above it at the previous row: 1, 4, 6, 4, and 1. These are the coefficients needed for any binomial expanded to the 5th power. It's a handy guide for quickly finding the coefficients without manually calculating combinations.
Binomial Theorem
The binomial theorem is a formula that provides a way to expand expressions of the form \((a+b)^n\). It states that \((a+b)^n\) expands into a sum involving terms of the form \({n \choose k}a^{n-k}b^k\), where \({n \choose k}\) are binomial coefficients. The power of this theorem comes from its ability to break down expressions into sums of terms with different powers. Using this theorem simplifies calculations when expanding binomials like \((2t - s)^5\), turning a potentially complex multiplication problem into a straightforward application of coefficients and powers.
Coefficients
Coefficients in a binomial expansion are the numbers that multiply the variables raised to powers in each term of the expression. In the expansion of \((2t - s)^5\), coefficients are given by the 5th row of Pascal鈥檚 Triangle: 1, 5, 10, 10, 5, and 1. These coefficients tell us how many times each term of the expansion will be multiplied by its identical variables. They effectively simplify the process of binomial expansion by indicating how to balance the powers of each component within the expression. So when calculating, the coefficient helps in determining the specific multiplier of each term, making the algebraic expansion much clearer.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the context of binomial expansion, algebraic techniques help in simplifying expressions to their cleanest form. For example, when expanding \((2t - s)^5\), algebra involves substituting the variables and carrying out operations such as multiplication and addition. With algebra, the expanded form becomes precise, with organized powers of \(t\) and \(s\). This helps in solving equations, understanding patterns, and conducting accurate calculations, which are essential skills in mathematical reasoning.

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