1. To find the solutions of the polynomial equation x^4 - 52x^2 + 576 = 0, we can use substitution or factoring. Let's use factoring:
x^4 - 52x^2 + 576 = (x^2 - 36)(x^2 - 16) = 0
Now, set each factor equal to zero:
x^2 - 36 = 0 or x^2 - 16 = 0
Solving the first equation, x^2 - 36 = 0, we can factor it further:
(x - 6)(x + 6) = 0
Setting each factor equal to zero:
x - 6 = 0, x + 6 = 0
Solving for x gives us two real solutions: x = 6 and x = -6.
Solving the second equation, x^2 - 16 = 0, we can factor it further:
(x - 4)(x + 4) = 0
Setting each factor equal to zero:
x - 4 = 0, x + 4 = 0
Solving for x gives us two additional real solutions: x = 4 and x = -4.
Therefore, the real solutions of the polynomial equation x^4 - 52x^2 + 576 = 0 are 4, -4, 6, and -6.
Answer: C) 4, -4, 6, -6
2. To find the solutions of the polynomial equation x^3 = 216, we can use the cube root:
x = ā216
The cube root of 216 is 6, since 6 x 6 x 6 = 216.
Therefore, x = 6 is one real solution.
Now, let's find the imaginary solutions. We can rewrite the equation as:
(x - 6)(x^2 + 6x + 36) = 0
Setting each factor equal to zero:
x - 6 = 0
x^2 + 6x + 36 = 0
Solving the first equation, x - 6 = 0, gives us x = 6 as the real solution (which we already found).
Now, let's solve the second equation, x^2 + 6x + 36 = 0, using the quadratic formula:
x = (-b Ā± ā(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 6, and c = 36. Plugging these values into the quadratic formula, we get:
x = (-6 Ā± ā(6^2 - 4(1)(36))) / (2(1))
x = (-6 Ā± ā(36 - 144))/2
x = (-6 Ā± ā(-108))/2
x = (-6 Ā± ā(-36 * 3))/2
x = (-6 Ā± 6ā3i)/2
x = -3 Ā± 3ā3i
Therefore, the real or imaginary solutions of the polynomial equation x^3 = 216 are: -6, -3 + 3ā3i, and -3 - 3ā3i.
Answer: B) -6, 3 + 3ā3i, and 3 - 3ā3i.
3. To find the dimensions of the shipping box in inches, we need to solve for x in the equation:
(width)(length)(height) = 7.6 ft^3
Since 1 ft^3 = 1,728 in^3, we can convert 7.6 ft^3 to inches as follows:
7.6 ft^3 * 1,728 in^3/ft^3 = 13,123.2 in^3
Now, substitute the expressions for the dimensions in terms of x:
(x)(x + 5)(3x - 1) = 13,123.2
Expand the equation:
3x^3 + 11x^2 - 5x - 13,123.2 = 0
Unfortunately, this is a cubic equation, and solving it directly can be complex.
However, given the answer choices, we can iterate through the dimensions provided and calculate their volume in inches using the formula:
Volume = (width)(length)(height)
Checking each answer choice:
A) (15)(20)(44) = 13,200 in^3
B) (12)(17)(35) = 7,140 in^3
C) (15)(20)(45) = 13,500 in^3
D) (12)(17)(36) = 8,208 in^3
The closest answer choice to 13,123.2 in^3 is B) 12 in. by 17 in. by 35 in.
Answer: B) 12 in. by 17 in. by 35 in.