Answer :
Final Answer:
The polynomial f(x)=5x⁶+7x⁴+x²+9 has no real zeros because its discriminant is positive, indicating that it has six distinct real roots.
Explanation:
Polynomials have real zeros (roots) when they cross the x-axis on the Cartesian plane. To determine whether f(x) has real zeros, we can look at its discriminant, which is a mathematical expression that gives information about the nature of the roots.
In this case, the discriminant is related to the coefficients of the polynomial and is defined as Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c. For the polynomial f(x)=5x⁶+7x⁴+x²+9, we can consider it as a quadratic equation in terms of x². Therefore, a = 5, b = 7, and c = 1.
Δ = 7² - 4 * 5 * 1 = 49 - 20 = 29.
The discriminant, Δ, is positive (Δ > 0). When the discriminant is positive, it indicates that the quadratic equation has two distinct real roots. In this case, it means that the polynomial f(x) has real zeros for x², and since the highest power in f(x) is x⁶, it implies that f(x) has six real zeros.
However, f(x) has an additional term, which is a constant (9). Since constants don't affect the real roots of the polynomial, the presence of 9 does not change the conclusion.
Learn more about Polynomial discriminants
brainly.com/question/12355760
#SPJ11
