If f(x)=(x-3)^9+(x-3^2)^4+….+(x-3^4)^5, then (A) f(x) is always increasing (B) f(x)=0 has one re

step 1 So in this question we have f of x is equal to x minus 3 raise to the power 9 plus x minus 3 squ

If f(x)=(x-3)^9+(x-3^2)^4+….+(x-3^4)^5, then (A) f(x) is...

If $f(x)=(x-3)^{9}+\left(x-3^{2}\right)^{4}+\ldots .+\left(x-3^{4}\right)^{5}$, then
(A) $f(x)$ is always increasing
(B) $f(x)=0$ has one real $\&$ eight imaginary roots
(C) $x=3,3^{2} \quad 3^{9}$ are the roots of $f(x)=0$
(D) $f(x)=0$ has a negative real root

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has as many roots in the complex number system (counting multiplicities) as its degree. This is crucial in determining the possible number of real versus complex (imaginary) roots a polynomial might have.

Real vs Complex Roots

In polynomial equations, the real roots correspond to x-values where the graph crosses or touches the x-axis, while complex roots occur in conjugate pairs when the polynomial does not have x-intercepts corresponding to those roots. Understanding the distinction is key when analyzing the full solution set of polynomial equations.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of a variable with constant coefficients. They are significant because their behavior—such as end behavior, degree, and number of turning points—is determined by their degree and coefficients, and they form the basis for many problems in algebra and calculus.

Monotonicity

Monotonicity refers to functions that are either entirely non-decreasing or non-increasing over their domain. An increasing function is one where a larger input always yields a larger output. This concept is useful for understanding the behavior of functions without needing detailed calculations of their values everywhere.

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