Final answer:
The signs in a binomial expansion alternate only if the binomial includes a subtraction (a - b). For a binomial (a + b), all terms in the expansion are positive. The alternation of signs occurs due to the powers applied to the negative component in the binomial.
Step-by-step explanation:
The student asked whether binomial expansion alternates signs. In Mathematics, the binomial expansion of an expression (a + b)^n is represented by a series, which is given by the binomial theorem. According to this theorem, the signs of the terms in the expansion depend on the exponents and whether the original binomial includes subtraction or addition.
When expanding a binomial of the form (a + b)^n, where both a and b are positive and n is a non-negative integer, all the terms of the expansion are positive due to the nature of the exponents. However, if the binomial is (a - b)^n, the signs will alternate, starting with a positive sign. This is because the negative sign in the binomial gets raised to successive powers, which is equivalent to flipping the sign of the subsequent terms of the series expansion.
An example of such an expression would be (a - b)^3, which expands to a^3 - 3a^2b + 3ab^2 - b^3, where the signs clearly alternate with each term. In cases where the power is even, for example (a - b)^2, the expanded form is a^2 - 2ab + b^2, and we can see that the sign changes from positive to negative, then back to positive.