The coefficient of x in the expansion of f(x)=| (1+x)^21 (1+x)^22 (1+x)^a3 (1+x)^

The coefficient of x in the expansion of f(x)=| (1+x)^21...

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a scalar value that summarizes certain properties of the matrix, such as invertibility and volume scaling under the associated linear transformation. It is computed via specific rules (such as Laplace expansion) and has the key property of being multilinear with respect to the rows and columns of the matrix.

Binomial Expansion

The binomial expansion refers to the process of expanding expressions of the form (1+x)^n using the binomial theorem. In this expansion, the coefficient of the term x is given by n, which is fundamental when determining the contributions of linear terms in polynomial expressions.

Multilinearity of Determinants

One of the crucial properties of determinants is that they are linear in each row (or column) separately. This means that when each entry in a row is a sum of constants and terms involving x, the determinant's expansion in powers of x can be understood as the sum of determinants where, in each term, exactly one row contributes the x-term while the other rows contribute their constant parts.

Coefficient Extraction in Polynomial Expansions

Coefficient extraction is the process of identifying the coefficient of a specific term (such as the linear term in x) in a polynomial expansion. This typically involves expanding each term, selecting those parts that will combine to form the desired power, and summing their contributions appropriately.

Transcript

00:01 In this problem, we would like to evaluate the coefficient of x and this function f of x is given in the determinant form and make it note that a 1 a 2 a 3 b1 b2 b3 c1 c2c 3 are constants some real numbers all right the only variable that is existing in this particular thing is x because it's a function of x only right now how to evaluate this coefficient of x this is the main aim of this problem is coefficient of x in f of x.

00:35 Basically when you expand this determinant, it will be in the polynomial form.

00:40 But expanding this determinant is a very tedious task.

00:44 So what we need to do is we will actually use a binomial theorem.

00:49 We know from standard binomial theorem, 1 plus x whole power n is 1 plus nx plus n into n minus 1 by 2 into x square and some extra terms but do we need this term because as far as this problem is concerned we are only interested in coefficient of x so obviously when you have x square term it will dominate the power of 1 because the power of x in coefficient of x is 1 so we don't need these terms so what we do is we'll truncate we'll actually ignore these particular terms that is the terms greater than the power of 2 greater than it equal to the power of 2 so that will simply the problem.

01:30 So, what we do now is we'll write f of x as, so f of x will be determinant of by ignoring the order of degree two terms and above, we'll get 1 plus a 1x and then 1 plus what is the yeah, 1 plus a 2x 1 plus a 2x 1 plus a 2x and here will get 1 plus b1 x 1 plus b2 x and here 1 plus b3x and here 1 plus c1 x here 1 plus c2 x and here 1 plus c2 x and here 1 plus c3 x so this is a very simple determinant we can apply the elementary row operations to simplify this determinant yes so let's apply so we can apply something like this the row 2 is row 2 minus row 1 and row 3 is row 3 minus row 2 so let's apply these two elementary operations so then we get f of x is equal to the first row will not change because we are not disturbing that row so 1 plus a 1x 1 plus a 2 x 1 plus a 3 x and now let's go to the second row.

02:30 So in the second row, when you subtract this minus this, it will be b1 minus a1 into x.

02:36 So it will be b1 minus a1 into x and this will be b2 minus a2 into x and this will be b3 minus a3 into x.

02:46 Sorry, b3 minus a3 into x and likewise this will be c1 minus a1 into x and this will be c2 minus a2 into x...

Please give Ace some feedback