College

Use the Binomial Theorem to expand \((x - 2)^6\).

A. \(x^6 - 2x^5 + 4x^4 - 8x^3 + 16x^2 - 32x + 64\)

B. \(x^6 - 12x^5 + 24x^4 - 36x^3 + 48x^2 - 60x + 12\)

C. \(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)

D. \(x^6 - 32x^5 + 16x^4 - 8x^3 + 4x^2 - 2x + 64\)

Answer :

Answer: C

Step-by-step explanation:

trust

Using Binomial Theorem To Expand

Use this formula: (x − y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + C(n, n-1) * x^1 * y^(n-1) + C(n, n) * x^0 * y^n

where C(n, k) represents the binomial coefficient "n choose k".

In this case, we have (x − 2)^6, so the binomial coefficients will come from Pascal's Triangle:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

Using the formula, we can expand (x − 2)^6 as follows:

C(6, 0) * x^6 * (-2)^0 + C(6, 1) * x^5 * (-2)^1 + C(6, 2) * x^4 * (-2)^2 + C(6, 3) * x^3 * (-2)^3 + C(6, 4) * x^2 * (-2)^4 + C(6, 5) * x^1 * (-2)^5 + C(6, 6) * x^0 * (-2)^6

Simplifying each term and combining like terms, we get:

1 * x^6 * 1 + 6 * x^5 * (-2) + 15 * x^4 * 4 + 20 * x^3 * (-8) + 15 * x^2 * 16 + 6 * x^1 * (-32) + 1 * x^0 * 64

Which further simplifies to:

x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64

Therefore, the correct expansion of (x − 2)^6 is:

C. x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64

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