Question

binomial theorem (pascals triangle)
score: 2/5 penalty: none
question
use pascals triangle to expand (3 - y)^4. express your answer in simplest form.
answer attempt 1 out of 2

Explanation:

Step1: Recall Pascal's Triangle coefficients for power 4

The coefficients for $(a + b)^n$ when $n = 4$ from Pascal's Triangle are 1, 4, 6, 4, 1.

Step2: Identify a and b in the binomial

Here $a=3$ and $b=-y$.

Step3: Apply the binomial expansion formula $(a + b)^n=\sum_{k = 0}^{n}C(n,k)a^{n - k}b^{k}$

$(3 - y)^4=1\times3^{4}\times(-y)^{0}+4\times3^{3}\times(-y)^{1}+6\times3^{2}\times(-y)^{2}+4\times3^{1}\times(-y)^{3}+1\times3^{0}\times(-y)^{4}$

Step4: Calculate each term

$3^{4}=81$, $3^{3}=27$, $3^{2}=9$, $3^{1}=3$, $3^{0}=1$.
The terms are:

• $1\times81\times1 = 81$
• $4\times27\times(-y)=-108y$
• $6\times9\times y^{2}=54y^{2}$
• $4\times3\times(-y^{3})=-12y^{3}$
• $1\times1\times y^{4}=y^{4}$

Step5: Combine the terms

$(3 - y)^4=y^{4}-12y^{3}+54y^{2}-108y + 81$

Answer:

$y^{4}-12y^{3}+54y^{2}-108y + 81$