Question

Factor the expression 3sin²(x) - 5sin(x) - 2.

A. (3sin(x) + 1)(sin(x) - 2)
B. (3sin(x) - 1)(sin(x) + 2)
C. (3sin(x) - 2)(sin(x) + 1)
D. (3sin(x) + 2)(sin(x) - 1)

Answer 1

The expression 3sin²(x) - 5sin(x) - 2 is factored into (3sin(x) + 1)(sin(x) - 2) by finding two numbers that multiply to -6 and add to -5, which are -6 and +1.

Step-by-step explanation:

To factor the expression 3sin²(x) - 5sin(x) - 2, we will look for two numbers that multiply to give the product of the coefficient of the sin²(x) term and the constant term (-2 * 3 = -6) and add up to the coefficient of the sin(x) term (-5).

The two numbers that satisfy these conditions are -6 and +1, because (-6) * (+1) = -6 and (-6) + (+1) = -5. Rewriting the expression with these two numbers, we have:

3sin²(x) - 6sin(x) + sin(x) - 2

Now we can factor by grouping:

(3sin²(x) - 6sin(x)) + (sin(x) - 2)

3sin(x)(sin(x) - 2) + 1(sin(x) - 2)

Now factor out the common factor (sin(x) - 2):

(3sin(x) + 1)(sin(x) - 2)

So the correct factored form of the expression is (3sin(x) + 1)(sin(x) - 2).