Question

I need help with both preferably with the Pascal’s Triangle.

Answer 1

Check the picture below.

as you already know, the first expanded terms start off with a highest exponent, and gradually go down, likewise, the second expanded terms start off with 0 and gradually go to the highest exponent.

the Pascal triangle simply gives us the coefficient both of those expanded terms multiply by.

Answer 2

(x+ 5)⁶ = x⁶ + 30x⁵ + 375x⁴ + 2500x³ + 9375x² + 18 750 + 15 625

(4x + 3)⁵ = 1024x⁵ + 3840x⁴ + 5760x³ + 4320x² + 1620x + 243

Explanation:

Pascal's Triangle gives us the coefficients of the terms in the binomial expansion of (x+ y)ⁿ.

1. (x + 5)⁶

The exponent is 6, so we use the coefficients corresponding to that exponent:

1, 6, 15, 20, 15, 6, 1.

Treat the 5 as if it were a single term. Then,

(x+ 5)⁶

= 1x⁶ + 6x⁵(5)¹ + 15x⁴(5)² + 20x³(5)³ + 15x²(5)⁴ + 6x¹(5⁵) + (5)⁶

= x⁶ + 30x⁵ + 375x⁴ + 2500x³ + 9375x² + 18 750 + 15 625

2. (4x + 3)⁵

The coefficients are

1, 5, 10, 10, 5, 1.

(4x + 3)⁵

= (4x)⁵ + 5(4x)⁴(3)¹ + 10(4x)³(3)² + 10(4x)²(3)³ + 5(4x)¹(3)⁴ + (3)⁵

= 1024x⁵ + 3840x⁴ + 5760x³ + 4320x² + 1620x + 243