Factor completely the polynomial. Show detailed solution.
What is the solution if
h^8 + 8h^4+ 16
it the equation and,
(h²-2h+2)²(h²+2h+2)²
is the answer.
(Note that if you get the answer h⁴+4 it is not yet prime so it can still be factored)
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Answer:
Let's factor the polynomial h^8 + 8h^4 + 16 step by step.
1. Start with the given polynomial: h^8 + 8h^4 + 16.
2. Recognize that this polynomial is a perfect square trinomial, which means it can be factored as the square of a binomial.
3. We can rewrite the polynomial as (h^4)^2 + 2 * h^4 * 2 + 2^2.
4. Now, notice that this expression fits the pattern of (a + b)^2 = a^2 + 2ab + b^2, where a = h^4 and b = 2.
5. Apply the pattern to our expression: (h^4 + 2)^2.
So, the completely factored form of the polynomial h^8 + 8h^4 + 16 is (h^4 + 2)^2.
Now, let's check if it matches the provided answer h^4 + 4. If we square (h^4 + 2), we get:
(h^4 + 2)^2 = h^8 + 4h^4 + 4.
This is not the same as h^8 + 8h^4 + 16, so the given answer h^4 + 4 is not correct.
The correct completely factored form of h^8 + 8h^4 + 16 is (h^4 + 2)^2.
The main answer to the factoring of the polynomial h^8 + 8h^4 + 16 is:
(h^4 + 2)^2