Question

binomial theorem (pascals triangle)
this is the only question in this section.
question
use pascals triangle to expand $(2y^2 - 2)^3$. express your answer in simplest form.
answer attempt 1 out of 2

Explanation:

Step1: Recall Pascal's Triangle for exponent 3

For \((a + b)^n\), when \(n = 3\), the coefficients from Pascal's Triangle are \(1, 3, 3, 1\). The binomial expansion formula is \((a + b)^3=\binom{3}{0}a^3b^0+\binom{3}{1}a^2b^1+\binom{3}{2}a^1b^2+\binom{3}{3}a^0b^3 = a^3 + 3a^2b+3ab^2 + b^3\).

Step2: Identify \(a\) and \(b\) in \((2y^2-2)^3\)

Let \(a = 2y^2\) and \(b=- 2\).

Step3: Substitute into the expansion formula

\[

$$\begin{align*} (2y^2-2)^3&=(2y^2)^3+3(2y^2)^2(-2)+3(2y^2)(-2)^2+(-2)^3\\ &=8y^6+3\times4y^4\times(-2)+3\times2y^2\times4+(-8)\\ &=8y^6 - 24y^4+24y^2 - 8 \end{align*}$$

\]

Answer:

\(8y^6 - 24y^4 + 24y^2 - 8\)