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Chapter 14: Problem 28

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-3 y)^{5}$$

Short Answer

Expert verified
The expanded form of the given binomial expression \( (x - 3y)^5 \) is \( x^5 - 15x^4y + 90x^3y^2 - 270x^2y^3 + 405xy^4 - 243y^5 \)

Step by step solution

01

Understand the Binomial Theorem

The Binomial Theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} {n\choose k} a^{n - k} * b^k \] where: \n- \( n \) is the power to which the binomial is raised. \n- \({n\choose k}\) denotes the binomial coefficient 'n choose k'. \n- \(a^{n - k}\) is the first term raised to the 'n - k'. \n- \(b^k\) is the second term raised to the 'k'.
02

Substitute the values into the Binomial Theorem

For our exercise, a is 'x', b is '-3y' and n is 5. Therefore, \[ (x - 3y)^5 = \sum_{k=0}^{5} {5\choose k} x^{5 - k} * (-3y)^k \]
03

Expand the binomial

By substituting the values into the binomial expansion equation, the simplified form is: \[ (x - 3y)^5 = {5\choose 0} x^{5} * (-3y)^0 + {5\choose 1} x^{4} * (-3y)^1 + {5\choose 2} x^{3} * (-3y)^2 + {5\choose 3} x^{2} * (-3y)^3 + {5\choose 4} x^{1} * (-3y)^4 + {5\choose 5} x^{0} * (-3y)^5 \]
04

Calculate the coefficients and simplify

Calculating the binomial coefficients and reducing the expression gives the expanded form: \[ x^5 - 15x^4y + 90x^3y^2 - 270x^2y^3 + 405xy^4 - 243y^5 \]

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