Question

pascals triangle quiz
algebra ii b / module 7: sequences and series

1. use pascals triangle to expand the binomial $(3x - 4)^3$.

$27x^3 + 108x^2 + 144x + 64$
$27x^3 - 108x^2 + 144x - 64$
$27x^3 - 108x^2 - 144x + 64$
$27x^3 + 108x^2 + 144x - 64$

Explanation:

Step1: Get Pascal's coefficients for $n=3$

The coefficients from the 4th row of Pascal's Triangle are $1, 3, 3, 1$.

Step2: Apply binomial expansion formula

For $(a+b)^3 = 1a^3b^0 + 3a^2b^1 + 3a^1b^2 + 1a^0b^3$, substitute $a=3x$, $b=-4$.

$$\begin{align*} &1(3x)^3(-4)^0 + 3(3x)^2(-4)^1 + 3(3x)^1(-4)^2 + 1(3x)^0(-4)^3\\ \end{align*}$$

Step3: Calculate each term

First term: $1\cdot27x^3\cdot1 = 27x^3$
Second term: $3\cdot9x^2\cdot(-4) = -108x^2$
Third term: $3\cdot3x\cdot16 = 144x$
Fourth term: $1\cdot1\cdot(-64) = -64$

Step4: Combine all terms

$27x^3 - 108x^2 + 144x - 64$

Answer:

B. $27x^3 - 108x^2 + 144x - 64$