EXAMPLE PROBLEMS USING FACTOR THEOREM

About "Example Problems Using Factor Theorem"

Example Problems Using Factor Theorem :

Here we are going to see some example problems to understand factor theorem.

To know the steps in factor theorem, please visit the page "Solving determinants using factor theorem".

Example Problems Using Factor Theorem - Questions

Question 1 :

Solve the following problems by using Factor Theorem :

(1) Show that

Solution :

Let us apply a = 0

Let us factor c and b from column 2 and 3. After factor out b and c, C₂ and C₃ will be identical. So, the determinant will become 0.

Hence a is a factor. By applying the b = 0 and and c = 0, we get the same result.

So the factors are a, b and c.

Leading diagonals are (b + c), (c + a) and (a + b). The sum of the exponents of leading diagonal is 3.

m = 3 - 3  =  0

So the required factor will be constant (k).

 2 [4 - 0]  =  k

 k  =  8

Hence proved.

Question 2 :

Solve

Solution :

Let the given determinant as delta. By applying the value x = 0, we get a in the first column, b in the second column and c in the third column. After factoring a, b and c from the first, second and third column respectively, we get identical rows and columns.

So (x - 0)² is a factor. Since the given matrix is in cyclic symmetric form, we may apply (x + a + b + c).

So, x + a = -b - c 

x + b  =  -a - c

and x + c  =  -a -b

If one column in the determinant is 0, its determinant value will become 0.

x²  (x + a + b + c)  =  0

x  =  0, 0, -(a + b + c)

Question 3 :

Show that

Solution :

By applying a = b, we get

First and second rows are equal. Determinant will become 0. So (a - b) is a factor.

By applying b = c, we get 2 identical rows. By applying c = a, we will get identical rows. So, (b - c) and (c - a) are factors.

So far we get 3 factors.

Sum of leading diagonals  =  4

m = 4 - 3  =  1

So, the required factor will be k(a + b + c).

5(18 - 12) - 1(36 - 12) + 1(12 - 6)  =  12k

5(6) - 1(24) + 1(6)  =  12k

30 - 24 + 6  =  12k

12k  =  12

k = 1

By applying the value of k, we get the proof.

After having gone through the stuff given above, we hope that the students would have understood, "Example Problems Using Factor Theorem". 

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