<p>If n is an integer, what is the remainder when 3x^(2n+2) -4x^(2n+2) +5x^(2n+1) -8 is divided b x+1
A) -20
B)-10
C)-4
D)0
correct choice is S but i can not understand the explanation.</p>
<p>If x coordinate system is devised so that the positive y-axis makes an angle of 60 with positive x-axis,what is the distance between the points with coordinates (4,-3) and (5,1)
A) 3.61
B)3.87
C) 4.12
D)4.58
E)7.14
Answer is D but once again i can not understand why</p>
<p>For the first question you have to apply the rule of division for polynomials. If the polynomial is divided by x+b, the remainder will be the value you get when you replace the x’s with -b. So here you have to replace the x’s with -1. The value of n dosent really matter, the only thing it does is telling us whether x will be raised at an odd or even number, hence whether the coefficient next to the x will take a positive or negative value.
Following this, the solution becomes 3<em>(-1)^(even number)-4(-1)^(even number)+5</em>(-1)^(odd number)-8 which equals -14</p>
<p>For the second problem the key is to find the horizontal distance between the two points, so you can form a right triangle with the vertical one and derive the distance with pythagoras. The vertical distance is 1-(-3)=4, we got one leg. For the horizontal leg you need to see that the difference of 1 between the x coordinates is the hypotenuse of a 30-60-90 triangle. The angle of 60 is pointing towards the horizontal distance of the two points, so the distance is sqrt(3)/2.
Then use pythagoras to find out that 4^2+sqrt(3)/2^2=x^2, so x=4.1</p>
<ol>
<li>(Bezout’s theorem) – the remainder when a polynomial f(x) is divided by x-a is equal to f(a).</li>
</ol>
<p>In this case, f(-1) = 3<em>(-1)^(2n+2) - 4</em>(-1)^(2n+2) + 5*(-1)^(2n+1) - 8
= 3 - 4 - 5 - 8 = -14.</p>
<ol>
<li>For this one I drew the x-axis horizontally and the y-axis at a 60-degree angle. Let A = (5,1) and B = (4,-3). The horizontal distance of AB is 1.</li>
</ol>
<p>Finding the vertical distance was a bit tricky. I let C = (5,-3), and D be the intersection of the lines x = 5 and the “horizontal” line through A. It’s pretty clear that BC = 8sqrt(3)/3. CD can be found using 30-60-90 triangles, in which CD = sqrt(3)/3.</p>
<p>Hence, the vertical distance BD = BC - CD = 7sqrt(3)/3, then AB = sqrt(1 + BD^2), which is roughly 4.16. Weird, both of us got C…and our answers don’t even match up.</p>