If the polynomials f(x)=x^2+5 x-p and g(x)=x^2-2 x+6 p have a common factor, then the common fac

step 1 So in the given question we are told that there is a function f of x which is given as x squared

If the polynomials f(x)=x^2+5 x-p and g(x)=x^2-2 x+6 p have...

If the polynomials $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+5 \mathrm{x}-\mathrm{p}$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}-2 \mathrm{x}+6 \mathrm{p}$ have a common factor, then the common
factor is
(1) $x+2$
(2) $\mathrm{x}$
(3) $x+4$
(4) Either (2) or (3)

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Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. This process is essential when analyzing polynomials for common factors, as it simplifies the expressions and reveals underlying relationships between the polynomials.

Roots and Shared Roots

The concept of roots refers to the values of x for which a given polynomial evaluates to zero. When two polynomials share a common factor, they have at least one common root. Identifying whether two polynomials share a common root is a crucial step in determining if they have a common factor.

Factor Theorem

The factor theorem is a fundamental result in algebra which states that a polynomial f(x) has a factor (x - r) if and only if f(r) = 0. This theorem provides a direct link between the roots of a polynomial and its factorization, making it a key tool for identifying common factors.

Common Factor in Polynomials

In algebra, a common factor of two polynomials is a nonconstant polynomial that divides both of them without leaving a remainder. When two polynomials have a common factor, it means they share at least one root, reflecting a structural relationship between the expressions.

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