Given f(x)=2 x^3-14 x^2+b x-3 with f(2)=0, g(x)=x^3+c x^2-8 x+30, with the zero x=3-i, and b and

step 1 Here in this question we have given a function f x as 2x q minus 14 x square plus bx minus 3 and

Given f(x)=2 x^3-14 x^2+b x-3 with f(2)=0, g(x)=x^3+c x^2-8...

Key Concepts

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by x - a, the remainder is f(a). This concept is useful for checking whether a given number is a root of the polynomial, as a zero remainder indicates that x - a is indeed a factor.

Factor Theorem

Closely related to the Remainder Theorem, the Factor Theorem establishes that if f(a) = 0 for a polynomial f(x), then x - a is a factor of the polynomial. This is crucial for determining factors of a polynomial and subsequently simplifying or factoring it completely.

Complex Conjugate Root Theorem

For polynomials with real coefficients, the Complex Conjugate Root Theorem assures that if a complex number is a root, then its complex conjugate must also be a root. This theorem is essential when dealing with non-real roots of polynomials and ensures that the polynomial's factors remain in real form.

Coefficient Comparison

Coefficient comparison involves equating the coefficients of corresponding powers of x in a polynomial identity to solve for unknown coefficients. This method is particularly useful when some values or roots of the polynomial are known, allowing one to determine missing parameters in the polynomial expression.

Polynomial Evaluation

Polynomial evaluation is the process of substituting a particular value for x in a polynomial and calculating the resulting output. This concept is fundamental in verifying the roots of a polynomial, using the Factor and Remainder Theorems, and determining products of polynomial evaluations.

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