Question

Expand the binomial. (x + δx)^2

a. x^2 + 2δx
b. x^2 - δx^2
c. x^2 + 2xδx + δx^2
d. x^2 - 2xδx + δx^2

Answer 1

The expansion of the binomial (x + δx)^2 using the binomial theorem is x^2 + 2xδx + δx^2, which corresponds to option c.

Step-by-step explanation:

The question involves the expansion of a binomial, specifically squaring the binomial (x + δx)^2. The correct expansion is obtained using the binomial theorem, which in this case for an exponent of 2 is simply the square of the first term, plus twice the product of the two terms, plus the square of the second term. Therefore, the correct expansion is x^2 + 2xδx + δx^2, which matches option c.

The binomial theorem states that the square of a binomial (a + b)^2 is a^2 + 2ab + b^2. Applying this to our case:

1. The square of the first term: x^2.
2. Twice the product of the two terms: 2xδx (series expansions often use this step).
3. The square of the second term: δx^2.

Combining these three parts gives us the expanded binomial: x^2 + 2xδx + δx^2.