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Find the degree, leading coefficient, and the maximum number of real zeros of the polynomial \( f(x) = x^7 - 2x^6 + 6x^4 + 5x^6 \).

1. Degree:
2. Leading Coefficient:
3. Maximum number of real zeros:

A. 7, 1, 7
B. 7, 1, 6
C. 6, 1, 7
D. 6, 1, 6

Answer :

Final answer:

The polynomial f(x) = x⁷ - 2x⁶ + 6x⁴ + 5x⁶ has a degree of 7, a leading coefficient of 1, and a maximum number of 7 real zeros, combining like terms and applying the Fundamental Theorem of Algebra and Descartes' Rule of Signs.

Explanation:

The question involves identifying the degree, leading coefficient, and the maximum number of real zeros of the polynomial f(x) = x⁷ - 2x⁶ + 6x⁴ + 5x⁶. First, we combine like terms, which in this case is x⁶, resulting in f(x) = x⁷ + 3x⁶ + 6x⁴. The highest power of x is 7, so the degree of the polynomial is 7.

The leading coefficient is the coefficient of the term with the highest power, which is 1 in this case. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex roots (counting multiplicities). However, in terms of real roots, Descartes' Rule of Signs tells us that the maximum number of real zeros of a polynomial is equal to its degree. Thus, the maximum number of real zeros for this polynomial is also 7.

The correct answer from the provided options is (a) 7, 1, 7.

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