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# fundamental operations on polynomials and factoring

A.Find p(x) + 3q(x)
p(x) = 2x^4 + 15x^3 - 10x^2 + 14
q(x) = 35x^4 - 16x^2 + x - 11

b.Find P(-1/2) if P(x) = 4x^4 -2x^3 + 17

c.Simplify: (-3x + 4x^2 + 6x^3) - (4 - x + 2x^3) - (-6x^2 + y)

d.Add: (10x^2 + 3y^2 - 6z^2 + xy + 4yz + 2xz) + (2x^2 - y^2 + 5z^2 - 2xy + 7yz - 3xz)

e.Multiply: (2x - 3y)^2

f.Multiply: (6x - 7) (6x + 7)

g.Divide: (3x^3 - x^2 + 10x - 4) ÷ (x + 3)

B. Factor completely:

a.x^2 - 5x - 7x + 35

b.2x^2 + 11x + 5

c.x^4y - 16y

C. Solve the following problems involving applications of polynomials.

a. A photo is 4 inches longer than it is wide. A 3-inch border is placed around the photo making the total area of the photo and border 165 square inches. What are the dimensions of the photo?

b. A rectangular patio is 7 ft longer than it is wide. Determine the dimensions of the patio if it measures 13 ft diagonally.

c. Three consecutive even integers are such that the square of the third is 100 more than the square of the second. Find the three integers.

I have completed most of these but was a little confused. I had the hardest problems with the word problems.

#### Solution Preview

A. Solve the following questions involving fundamental operations on polynomials
a.Find p(x) + 3q(x)
p(x) = 2x^4 + 15x^3 - 10x^2 + 14
q(x) = 35x^4 - 16x^2 + x - 11

Solution:
p(x) + 3q(x)
= (2x^4 + 15x^3 - 10x^2 + 14) + 3(35x^4 - 16x^2 + x - 11)
= 2x^4 + 15x^3 - 10x^2 + 14 + 105x^4 - 48x^2 + 3x - 33
= 107x^4 + 15x^3 - 58x^2 + 3x - 19

b.Find P(-1/2) if P(x) = 4x^4 -2x^3 + 17

P(-1/2)
= 4(-1/2)^4 - 2(-1/2)^3 + 17 = 4/16 + 2/8 + 17 = ¼ + ¼ + 17 = ½ + 17 = 17½ = 17.5

c. Simplify: (-3x + 4x^2 + 6x^3) - (4 - x + 2x^3) - (-6x^2 + y)

= -3x + 4x^2 + 6x^3 - 4 + x - 2x^3 + 6x^2 - y

= 4x^3 + 10x^2 -2x - y - 4