Let f(x)=| x^3+1 1 0 x^2-x -1 2 x^5+x^3+1 0 1 | Column I

step 1 In the problem, since mod A and mod B are the roots of p. square minus 2p plus 1, that equals 0,

Let f(x)=| x^3+1 1 0 x^2-x -1 2 ...

Let $f(x)=\left|\begin{array}{ccc}x^{3}+1 & 1 & 0 \\ x^{2}-x & -1 & 2 \\ x^{5}+x^{3}+1 & 0 & 1\end{array}\right|$ Column I Column II (a) constant term in $\mathrm{f}(\mathrm{x})$ is (p) $-1$ (b) coefficient of $\mathrm{x}$ in $\mathrm{f}(\mathrm{x})$ is (q) 1 (c) degree of the polynomial $\mathrm{f}(\mathrm{x})$ is (r) 12 (d) slope of the tangent to the curve $y=f(x)$ at $x=1$ is (s) 5

Let $f(x)=\left|\begin{array}{ccc}x^{3}+1 & 1 & 0 \\ x^{2}-x & -1 & 2 \\ x^{5}+x^{3}+1 & 0 & 1\end{array}\right|$
Column I Column II
(a) constant term in $\mathrm{f}(\mathrm{x})$ is
(p) $-1$
(b) coefficient of $\mathrm{x}$ in $\mathrm{f}(\mathrm{x})$ is
(q) 1
(c) degree of the polynomial $\mathrm{f}(\mathrm{x})$ is
(r) 12
(d) slope of the tangent to the curve $y=f(x)$ at $x=1$ is
(s) 5

Key Concepts

Determinant of a Matrix

This concept involves computing the determinant of a square matrix, which is a scalar value derived from its elements. Determinants provide useful information about the matrix, such as whether the matrix is invertible, and they are computed using specific formulas or expansion methods like cofactor expansion. When a determinant is derived from polynomial expressions, the resulting expression will be a polynomial whose properties, such as degree and coefficients, can be further analyzed.

Polynomial Functions and Coefficient Extraction

This concept deals with functions that are expressed in the form of polynomials. In problems where a polynomial is defined via the determinant of a matrix, one might need to extract specific information such as the constant term (the term independent of the variable) or the coefficient of a particular power of the variable. Understanding how to identify and extract these coefficients is crucial for analyzing and manipulating polynomial functions.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable present in the expression when the polynomial is written in its standard form. It provides a measure of the complexity and growth rate of the polynomial function. In contexts where a polynomial emerges from operations such as determinants, identifying its degree is essential for understanding its long-term behavior and for simplifying or approximating its form.

Differentiation of Polynomial Functions

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to the variable. For polynomial functions, differentiation is performed term by term, which allows for the straightforward computation of derivatives. The derivative is particularly important in finding the slope of the tangent line to a curve at a given point, thereby linking calculus techniques to the study of polynomial functions.

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