Problem 19 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (y-3)^{4} $$

Step-by-Step Solution

Answer

The binomial \( (y-3)^4 \) can be expanded and simplified to \( y^4 -12y^3 +54y^2 -108y + 81 \)

Step 1: Express the binomial expansion

The binomial \( (y-3)^{4} \) can be expanded using the Binomial theorem, which states that: \[ (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^{k}\] Using this theorem, the binomial is expressed as: \[ (y-3)^4= {4 \choose 0} y^4(-3)^0 + {4 \choose 1} y^3(-3)^1 + {4 \choose 2} y^2(-3)^2 + {4 \choose 3} y^1(-3)^3 + {4 \choose 4} y^0(-3)^4 \]

Step 2: Simplify the expression

The combinatorial terms \( {4 \choose 0}, {4 \choose 1}, {4 \choose 2}, {4 \choose 3}, {4 \choose 4} \) are evaluated as 1, 4, 6, 4, 1 respectively, and the powers of -3 as \( (-3)^0, (-3)^1, (-3)^2, (-3)^3, (-3)^4 \) are evaluated as 1, -3, 9, -27, 81 respectively. So the expression becomes: \[ 1y^4 + 4y^3(-3) + 6y^2(9) + 4y(-27) + 1(81) = y^4 -12y^3 +54y^2 -108y + 81 \]