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Algebra: Polynomials

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For the polynomial

P(x) = 4x2 + x + 3

1
a. What is the degree of P(x)?
b. What is the coefficient of the leading term?
c. What is the coefficient of the linear term?
d. What is the value of P(2), that is, the expression P(x) evaluated at x = 2?

Allowing Q(x) = x + 1,

e. What is the polynomial resulting from the arithmetic operation P(x) * Q(x)?

2
a. Exhibit the prime factorization of the integer 108.
b. Determine the greatest common factor of the set of integers { 28, 42, 56 }

For the polynomial

R(x) = 3x2 + x - 2

3.
a. Exhibit a factorization of the above polynomial R(x)
b. Determine the zeros of R(x).

For the polynomial

W(x) = x2 - 20x + 100

4.
a. Exhibit a factorization of the above polynomial W(x)
b. Determine the zeros of W(x).

For the polynomial

Y(x) = x2 + 2x -1

5.
a. Can the polynomial Y(x) be factored?
b. Determine the zeros of Y(x).
c. Verify that the zeros are valid by evaluating the expression for each zero.

6.
Simplify
a. See image002.gif attachment
b See image004.gif attachment

Perform the indicated operation. Reduce each answer to lowest terms.
7.
a. 1/(a - b) + 2/(a + b)
b. What is the domain of the rational expression of part a?

8.
a a/(a2 - 9) + 2a/(a - 3)
b What is the domain of the rational expression of part a?

9.

Well, here we are again with that English Channel, which appears to have repeated significance in flying 'firsts. Louis Bleriot, French aviator and inventor, was the first individual to successfully cross the English Channel in a motorized aircraft. On July 25, 1909, he flew his aircraft, called the Bleriot XI, a three cylinder, 25 horsepower monoplane, from Les Barraques, France to Dover, England, a distance of 22 miles, in 37 minutes. He received considerable acclaim and prize money. The Bleriot XI had a cruising speed of 47 miles/hr, At this cruising speed, the time to make the crossing would be determined as (T=D/R) t = 22 mi / (47 mi/hr) = .47 hours = 28 minutes. Thus, Louis must have experienced considerable headwind.

part i: Specify a rational expression which yields the flight time, Les Barraques to Dover, for the Bleriot XI as a function of headwind.
part ii: Calculate the headwind that Louis experienced on his successful flight

Note: The early inventions in powered flight, due to people like the Wright brothers, Curtis and Bleriot, must have been spiritual endeavors. An engineering team attempted to reproduce the first heavier than air flight by the Wright brothers for the hundredth anniversary in 2003 and the team could not accomplish it.
10. Civil War Artillery Piece

Part 1

We have a mortar which is fired at a 45 deg angle. It has an initial velocity component of 340 ft / sec in both the vertical as well as horizontal directions. We have, then, that our equation of height above ground as a function of time t is

h(t) = -16t2 + 340t

What is the ordered pair which constitutes the vertex of our parabolic equation h(t)?
How high will our projectile ascend? How long does the projectile take to gain that maximum height?

What are the ordered pairs corresponding to the zeros of our parabolic equation h(t)?
How long does it take for our projectile to reach its destination?

Part 2

Since our projectile travels at a constant rate of 340 ft / sec in the horizontal direction, we have, using the formula D = RT, that our horizontal distance x = 340 t, giving

t = x / 340

Our above equation h(t) can now be converted into h(x), as follows

See image012.gif attachment

Giving

See image014.gif attachment

What is the ordered pair which constitutes the vertex of our parabolic equation h(x)?
At what distance from our firing point does our projectile reach maximum height?

What are the ordered pairs corresponding to the zeros of the above equation h(x)?
How far does our projectile travel?