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Polynomial operations

Solve the following questions involving fundamental operations on polynomials
a. Find p(x) + 4q(x)
p(x)=4x^4 + 55x^3 - 23x^2 + 13
q(x)=43x^4+ 14x^2 -12
b. Find P(-1/2) if
P(x) = 2x^4 + x^3 + 12
c. Simplify: (-4 + x^2 + 2x^3) - (-6 - x + 3x^3) - (-6y^3 + y^2)
d. Add: (2x^2 + 6y^2 + 4z^2 + 3xy + yz + zx) + (4x^2 + 3y^2 + z^2 - 3xy - 9yz + 5zx)

e. Multiply: (3x + 3y)^2
f. Multiply: (3x + 4) (3x - 4)
g. Divide: (2x^3 - x^2 + 3x -1) รท (x + 2)
B. Factor completely:
a. x^2 - 7x - 9x + 63
b. 4x^2 + 34x + 42
c. x^8 - 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 108 square inches. What are the dimensions of the photo?
b. A rectangular parking lot is 100 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 500 ft diagonally.
c. Three consecutive odd integers are such that the square of the third is 264 more than the square of the second. Find the three integers.

Solution Preview

a)p(x) + 4q(x) = 4x^4 + 55x^3 - 23x^2 + 13 +4(43x^4+ 14x^2 -12)

= 4x^4 + 55x^3 - 23x^2 + 13 + 172x^4+56x^2-48

= 176x^4+ 55x^3+33x^2 - 35 <--- combine like terms.

b)P(-1/2) = 2(-1/2)^4 + (-1/2)^3 + 12 <-- replace x by -1/2

= 2(1/16) + (-1/8) + 12

= (1/8) + (-1/8) + 12

= 0 + 12 = 12

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

= -4 + x^2 + 2x^3 +6 + x - 3x^3 + 6y^3 - y^2 <--- remove "()"

= -x^3 + 6y^3 + x^2 - y^2 + x - 4 <---- this is final answer.

d. Add: (2x^2 + 6y^2 + 4z^2 + 3xy + yz + zx) + (4x^2 + 3y^2 + z^2 - 3xy - 9yz + 5zx)

= 2x^2 + 6y^2 + 4z^2 + 3xy + yz + zx + 4x^2 + 3y^2 + z^2 - 3xy - 9yz + 5zx

= 6x^2 + 9y^2 + 5z^2 - 8yz + 6zx <--- combine ...

Solution Summary

This provides several examples of working with polynomials, including factoring and fundamental operations.

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