This problem has two parts, Part A and Part B.
Part A: Expand (x+1)6 using the Binomial Theorem.
Part B: Which method(s) could help identify the correct answer to Part A?
Select one answer for Part A and select all answers that apply for Part B.
B: ∑6k=0(6k)x6−k1k=(60)x6−010+(61)x6−111+(62)x6−212+(63)x6−313+(64)x6−414+(65)x6−515+(66)x6−616
A: x6+6x5+15x4+20x3+15x2+6x+1
B: ∑6k=1(6k)x6−k1k=(61)x6−111+(62)x6−212+(63)x6−313+(64)x6−414+(65)x6−515+(66)x6−616
B: The row corresponding to (a+b)6 in Pascal’s Triangle has the terms 1, 6, 15, 20, 15, 6, and 1.
A: x6+5x5+15x4+25x3+15x2+5x+1
B: The row corresponding to (a+b)6 in Pascal’s Triangle has the terms 1, 5, 15, 25, 15, 6, and 1.
A: x6−6x5+15x4+20x3+15x2−6x−1
A: x6+6x5+10x4+25x3+10x2+6x+1
B: ∑6k=0(6k)x6−k1k=(60)x016−0+(61)x116−1+(62)x216−2+(63)x316−3+(64)x416−4+(65)x516−5+(66)x616−6
B: The row corresponding to (a+b)6 in Pascal’s Triangle has the terms 1, 6, 10, 25, 10, 6, and 1.
Hard to read all those choices
in effect, (x+1)^6
= x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1
the row in Pascal's triangle I used was 1 6 15 20 15 6 1
I also see (a+b)^6 , which would be
= a^6 + 6a^5 b + 15a^4 b^2 + 20a^3 b^3 + 15a^2 b^4 + 6a b^5 + b^6
Part A: To expand (x+1)^6 using the Binomial Theorem, we can use the formula:
(x+a)^n = C(n,0)a^n(x^0) + C(n,1)a^(n-1)(x^1) + C(n,2)a^(n-2)(x^2) + ... + C(n,n)a^0(x^n)
where C(n,k) represents the binomial coefficient, which is equal to n! / (k!(n-k)!).
In this case, a is 1, since we have (x+1). Also, n is equal to 6.
Expanding (x+1)^6 using the Binomial Theorem:
(x+1)^6 = C(6,0)(1^6)(x^0) + C(6,1)(1^5)(x^1) + C(6,2)(1^4)(x^2) + C(6,3)(1^3)(x^3) + C(6,4)(1^2)(x^4) + C(6,5)(1^1)(x^5) + C(6,6)(1^0)(x^6)
Simplifying, we get:
Part A: x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1
Now let's move on to Part B.
Part B: To identify the correct answer to Part A, we can consider a few methods:
1. Applying the Binomial Theorem: From the step-by-step expansion of (x+1)^6 using the Binomial Theorem, we can compare the given options to the expression we obtained. Only the option "A: x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1" matches the expanded form we found in Part A, so it is the correct answer.
2. Checking the coefficients: Another method is to observe the coefficients in the expanded form. The coefficients in the expanded expression relate to the binomial coefficients in Pascal's Triangle. By comparing the options to the corresponding row in Pascal's Triangle, we can identify which options have the correct coefficients. In this case, the correct row is the one with the terms 1, 6, 15, 20, 15, 6, and 1. Only option "B: The row corresponding to (a+b)^6 in Pascal’s Triangle has the terms 1, 6, 15, 20, 15, 6, and 1" matches this pattern.
Therefore, the correct answers are:
Part A: x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1
Part B: Option A and Option B