Question

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

Explanation:

Step1: Recall Pascal's Triangle 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 $a = 2x$ and $b=- 1$.

Step3: Apply the binomial - expansion formula

$(2x-1)^4=1\times(2x)^4\times(-1)^0 + 4\times(2x)^3\times(-1)^1+6\times(2x)^2\times(-1)^2 + 4\times(2x)^1\times(-1)^3+1\times(2x)^0\times(-1)^4$.

Step4: Simplify each term

• For the first term: $1\times(2x)^4\times(-1)^0=16x^4$.
• For the second term: $4\times(2x)^3\times(-1)^1=4\times8x^3\times(-1)=-32x^3$.
• For the third term: $6\times(2x)^2\times(-1)^2=6\times4x^2\times1 = 24x^2$.
• For the fourth term: $4\times(2x)^1\times(-1)^3=4\times2x\times(-1)=-8x$.
• For the fifth term: $1\times(2x)^0\times(-1)^4=1\times1\times1 = 1$.

Step5: Combine the terms

$(2x - 1)^4=16x^4-32x^3 + 24x^2-8x + 1$.

Answer:

$16x^4-32x^3 + 24x^2-8x + 1$