I need help with both preferably with the Pascal’s Triangle.
2 answers:
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:
(x+ 5)⁶ = x⁶ + 30x⁵ + 375x⁴ + 2500x³ + 9375x² + 18 750 + 15 625
(4x + 3)⁵ = 1024x⁵ + 3840x⁴ + 5760x³ + 4320x² + 1620x + 243
Step-by-step 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
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