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 we have part A. So here, F of x being a polynomial function, therefore the consta

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 $x$ in $f(x)$ 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 $x$ in $f(x)$ 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

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable present in the expression when the polynomial is expressed in its standard form. Determining the degree helps in understanding the overall behavior of the function, such as its growth rate and the potential number of roots. When polynomials are generated from determinants, the degree is determined by summing the degrees of the terms in the products that form the leading term.

Matrix Determinant

A determinant is a scalar value derived from a square matrix that carries important information about the matrix, such as whether it is invertible. In this context, the determinant is used as a function of a variable, meaning that the entries of the matrix are polynomials in that variable. Understanding how to compute determinants, especially using techniques like cofactor expansion or row reduction, is crucial to forming and simplifying the resulting polynomial expression.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to nonnegative integer powers with corresponding coefficients. When a determinant has polynomial entries, the resulting determinant becomes a polynomial function in the variable. The structure of polynomials—having terms ordered by degree—allows one to identify the constant term, the coefficient of any specific power, and the overall degree of the polynomial.

Constant Term and Coefficient Extraction

Extracting specific coefficients, such as the constant term or the coefficient of a particular power, involves identifying the contributions of the terms in a polynomial that do not involve the variable, or match the desired power. This technique is essential when analyzing parts of a polynomial function to understand its behavior at certain values, or to simplify computations involving the polynomial.

Differentiation

Differentiation is the process of finding the derivative of a function, which gives the rate at which the function's value changes with respect to its variable. For polynomial functions, differentiation simplifies to reducing the exponent by one and multiplying by the original exponent. Evaluating the derivative at a specific point provides the slope of the tangent to the curve at that point, which is a key concept in understanding instantaneous rates of change in calculus.

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