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Answer :
Final answer:
The real zeros of the polynomial f(x) = x⁶ - 7x⁴ - 8x² are found by factoring and substitution, resulting in the real zeros x = √8 and x = -√8.
Explanation:
To find the real zeros of the polynomial f(x) = x⁶ - 7x⁴ - 8x², we first notice that each term has an x-squared factor which suggests the substitution y = x². This will convert the given polynomial into a quadratic equation y² - 7y - 8 = 0. We can then factor this equation to get (y - 8)(y + 1) = 0, giving us the solutions y = 8 and y = -1.
Since we made the substitution y = x², we now need to solve x² = 8 and x² = -1. The equation x² = 8 has two real solutions, x = √8 and x = -√8. However, the equation x² = -1 has no real solutions because the square of a real number cannot be negative.
Therefore, the real zeros of the polynomial are x = √8 and x = -√8, which can also be expressed as approximately x = 2.83 and x = -2.83 when using decimal approximations.
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