Tutor profile: Nick N.

A second degree polynomial of the form $$3x^2+Ax+B$$ where A and B are constants is equal to 0 when $$x=3$$ and $$x=\frac{-2}{3}$$. What are the values of A and B?

To solve this problem we will begin by representing our second degree polynomial as the product of two first degree polynomials. Based on the information we are given, we can write: $$(x-3)(x+\frac{2}{3})=0$$ We can expand using the FOIL method to get: $$x^2-2\frac13x-2=0$$ We're almost done, but our new polynomial's leading coefficient does not match the one we were given. We can simply multiply our equation by 3 to find: $$3x^2-7x-6=0$$ So if $$3x^2+Ax+B=0$$ when $$x=3$$ and $$x=\frac{-2}{3}$$, then A = -7 and B = -6.

What is the tangent of 45 degrees?

Here we can use our knowledge of right triangles to work out the answer without a calculator. In a right triangle the tangent of an angle is equal to $$\frac{\text{Opposite Side Length}}{\text{Adjacent Side Length}}$$. Our geometry knowledge tells us that the legs in a 45º - 45º - 90º triangle are of equal size, so we can use our ratio $$\frac{\text{Opposite Side Length}}{\text{Adjacent Side Length}}$$ to show that the tangent of 45º is equal to 1.

You are searching for artifacts on the ocean floor using a remotely operated submarine. You want to view a shipwreck that is 12,000 ft below the surface, and your submarine descends at a constant rate of 16 ft every 3 seconds. About how long will it take your submarine to reach the wreck, assuming it begins at the surface? A) 4 minutes B) 12 minutes C) 38 minutes D) 46 minutes

There are multiple approaches you could use to solve this problem. One method would be to divide 16 ft by 3 seconds, getting a rate of 5 1/3 ft per second. To find the time it would take to travel 12,000 ft at this rate we divide 12,000 ft by 5 1/3 ft per second and get 2,250 seconds, which can be converted to 37.5 minutes. You could also reach this answer by setting $$\frac{16 ft}{3 seconds}$$ = $$\frac{12,000 ft}{X seconds}$$ and using cross products. To solve for X you would multiply 3 seconds by 12,000 ft to get 36,000 ft * seconds, which you then divide by 16 ft to get 2,250 seconds. The answer is C!