Question

1. use pascals triangle to expand the binomial \\((2x + 2)^4\\).

\\(16x^4 + 64x^3 + 96x^2 + 64x + 16\\)

\\(16x^4 - 64x^3 + 96x^2 - 64x + 16\\)

\\(16x^4 + 64x^3 + 96x^2 - 64x - 16\\)

\\(16x^4 - 64x^3 - 96x^2 + 64x - 16\\)

Explanation:

🆕 New Concept Discovered: Binomial Expansion with Pascal's Triangle
Using triangle patterns to expand powers of binomials.

Step 1: Identify Pascal's Triangle Coefficients

For a binomial raised to the power of \( n = 4 \), we look at the 4th row of Pascal's Triangle (where the second number is 4):

$$ 1, \quad 4, \quad 6, \quad 4, \quad 1 $$

These values serve as the coefficients for each term in our expansion.

Step 2: Set up the Binomial Theorem Formula

The expansion of \( (a + b)^4 \) is given by:

$$ (a + b)^4 = 1a^4b^0 + 4a^3b^1 + 6a^2b^2 + 4a^1b^3 + 1a^0b^4 $$

For our expression \( (2x + 2)^4 \), we substitute:

• \( a = 2x \)
• \( b = 2 \)

Step 3: Expand and Simplify Each Term

Now, we calculate each term individually:

• First term:

$$ 1 \cdot (2x)^4 \cdot 2^0 = 1 \cdot 16x^4 \cdot 1 = 16x^4 $$

• Second term:

$$ 4 \cdot (2x)^3 \cdot 2^1 = 4 \cdot 8x^3 \cdot 2 = 64x^3 $$

• Third term:

$$ 6 \cdot (2x)^2 \cdot 2^2 = 6 \cdot 4x^2 \cdot 4 = 96x^2 $$

• Fourth term:

$$ 4 \cdot (2x)^1 \cdot 2^3 = 4 \cdot 2x \cdot 8 = 64x $$

• Fifth term:

$$ 1 \cdot (2x)^0 \cdot 2^4 = 1 \cdot 1 \cdot 16 = 16 $$

Step 4: Combine the Terms

Combining all the simplified terms gives:

$$ 16x^4 + 64x^3 + 96x^2 + 64x + 16 $$

Since all terms in the original binomial \( (2x + 2) \) are positive, all terms in the expanded polynomial must also be positive.

Answer:

\( 16x^4 + 64x^3 + 96x^2 + 64x + 16 \) (First option)