Factor completely the polynomial. Show detailed solution.
h^8 + 8h^4+ 16
(Note that if you get the answer h⁴+4 it is not yet prime so it can still be factored)
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Factor completely the polynomial. Show detailed solution.
h^8 + 8h^4+ 16
(Note that if you get the answer h⁴+4 it is not yet prime so it can still be factored)
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Step-by-step explanation:
To factor the given polynomial, let's use a substitution. Let x = h^4.
Now we have the equation:
x^2 + 8x + 16
We can factor this quadratic equation using the quadratic formula or by completing the square. Let's use the quadratic formula.
The quadratic formula states:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 8, and c = 16. Plugging these values into the quadratic formula, we have:
x = (-8 ± √(8^2 - 4(1)(16))) / (2(1))
= (-8 ± √(64 - 64)) / 2
= (-8 ± √0) / 2
= (-8 ± 0) / 2
= -8 / 2
= -4
Since x = -4, we substitute back x = h^4:
h^4 = -4
To solve for h, we take the fourth root of both sides:
h = ±√(-4)
However, there are no real solutions for the square root of a negative number. Therefore, there are no real roots for the polynomial h^8 + 8h^4 + 16, and we cannot factor it further.
Thus, the polynomial h^8 + 8h^4 + 16 cannot be factored completely over the set of real numbers.