A. Solve the following questions involving fundamental operations on polynomials
A .Find p(x) + 4q(x) given p(x)= 4x4 + 10x3 - 2x2 + 13 and q(x) = 2x4+ 5x2 - 3
p(x) + 4q(x) = 4x4 + 10x3 - 2x2 + 13 + 8x4+ 20x2 – 12 = 12x4 + 10x3 + 18x2 + 1
b. Find P(-1/2) if P(x) = 2x4 + x3 + 12
P(-1/2) = 2(-1/2)4 + (-1/2)3 + 12 = 12
c. Simplify: (-4 + x2 + 2x3) – (-6 – x + 3x3) – (-6y3 + y2)
= -4 + x2 + 2x3 + 6 + x - 3x3 + 6y3 - y2 = 2 + x + x2 - x3 + 6y3 - y2
d. Add: (2x2 + 6y2 + 4z2 + 3xy + yz + zx) + (4x2 + 3y2 + z2 - 3xy - 9yz + 5zx)
= 2x2 + 6y2 + 4z2 + 3xy + yz + zx + 4x2 + 3y2 + z2 - 3xy - 9yz + 5zx
= 6x2 + 9y2 + 5z2 - 8yz + 6zx
e. Multiply: (3x + 3y)2 = 9x2 + 18xy + 9y2
f. Multiply: (3x + 4) (3x - 4) = 9x2 - 16
g. Divide: (2x3 - x2 + 3x -1) ÷ (x + 2)
2x3 - x2 + 3x -1 = 2x2(x-2) + 3x(x-2) + 9(x-2) + 17 = (x-2)(2x2 + 3x + 9) + 17
When divided by x-2, the quotient is (2x2 + 3x + 9) and the remainder is 17.
B. Factor completely:
a. x2 – 7x – 9x + 63 = (x-7)(x-9)
b. 4x2 + 34x + 42 = 2(2x + 3)(x + 7)
c. x4 - 1 = (x2 + 1)(x-1)(x+1)
C. Solve the following problems involving applications of polynomials.
a. A photo is 3 inches longer that it is wide. A 2-inch border is placed around the photo making the total area of the photo and border 108in2. What are the dimensions of the photo?
Let width be x, then length is x + 3.
Area after placing the border = (x + 4)(x + 7) = 108
Or x2 + 11x + 28 = 108
Or x2 + 11x -80 = 0
Or x2 + 16x – 5x - 80 = 0
Or x(x + 16) – 5 (x + 16) = 0
Or (x + 16)(x – 5) = 0.
Clearly width = x = 5 inches, and length = 8 inches.
b. A rectangular parking lot is 50 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 250 ft diagonally.
Let x be the width, then length is 50 + x.
Diagonal2 = length2 + width2
Or 2502 = (50 + x)2 + x2
Or x2 + x2 + 100x + 2500 = 62500
Or 2x2 + 100x – 60,000 = 0
Or x2 + 50x – 30,000 = 0
Therefore, x = [pic](taking the positive sign).
Therefore, width...