Question

"The Binomial Theorem

Use the Binomial Theorem to expand each power of a binomial.

1. (x - 5)4 =
2. (3x + 4) =
3. (2x - 3) =
4. (3x - y)4 =
5. (4x + 3y) ="

Answer 1

The Binomial Theorem allows for the expansion of a binomial raised to a power by summing up terms involving binomial coefficients and the binomial's terms raised to varying powers. Specific exponents must be known to apply the theorem fully, except for the provided example of
`(x - 5)^4`.

Step-by-step explanation:

The Binomial Theorem is a method used to expand expressions raised to a power, particularly when dealing with binomials. It states that a binomial raised to any power can be expanded as a sum of terms involving coefficients that arise from combinatorial calculations (binomial coefficients), each with a product of the binomial's terms raised to strategic powers. The general form of the Binomial Theorem for expanding
`(a + b)^n is a^n + n x a^(n-1)b + n(n-1)/2! x a^(n-2)b^2 + ... + b^n`, where each term involves the binomial coefficients.

The series expansions provided give a basis for applying the Binomial Theorem to the expressions at hand. To expand
`(x - 5)^4`, one would calculate each term by using the formula, taking into account the signs, which alternate because of the subtraction. However, the other examples
`(3x + 4)^n`,
`(2x - 3)^n`, and
`(4x + 3y)^n` do not have their exponents specified and would require further clarification for exact expansion. When expanding polynomials like these, remember that an exponent indicates repeated multiplication, which is relevant for understanding the underlying mechanics of binomial expansions.