If Δ(x)=| x 1+x^2 x^3 log(1+x^2) e^x sin x cos x tan x sin ^2 x

step 1 In problem 30 we have the determinant as a function of x equals x 1 plus x squared x cubed log o

If Δ(x)=| x 1+x^2 x^3 log(1+x^2) e^x ...

If $\Delta(x)=\left|\begin{array}{ccc}x & 1+x^{2} & x^{3} \\ \log \left(1+x^{2}\right) & e^{x} & \sin x \\ \cos x & \tan x & \sin ^{2} x\end{array}\right|$ then
(A) $\Delta(x)$ is divisible by $x$
(B) $\Delta(x)=0$
(C) $\Delta^{\prime}(x)=0$
(D) None of these

Key Concepts

Polynomial Divisibility

When a function, particularly one defined by a determinant, is said to be divisible by a polynomial such as x, it means that the polynomial is a factor of the function. This implies that the function will equal zero when the variable takes on the roots of the polynomial, which is a common concept in algebra to analyze the structure of a function.

Matrix Determinants

A determinant is a scalar value computed from the elements of a square matrix. It provides critical information about the matrix, such as whether it is invertible. In many problems, the structure of the determinant can reveal relationships among the matrix elements, such as linear dependence or special factoring properties.

Linear Dependence

Linear dependence occurs when one or more rows (or columns) of a matrix can be expressed as a combination of others. If a set of rows or columns is linearly dependent, the determinant of the matrix is zero. This concept is essential in verifying whether a determinant vanishes for all values of the variable.

Differentiation of Determinants

Differentiation of a determinant-valued function involves applying the standard rules of differentiation to the elements of the matrix. This might include the use of the chain rule, product rule, and other basic principles of calculus. Understanding how the determinant changes with respect to its variable is key to analyzing properties like the derivative equating to zero.

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