Factor Theorem Examples and Solutions :
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".
Question 1 :
Solve the following problems by using Factor Theorem :
(1) Solve
Solution :
By applying x = 0, we get identical rows and columns.
Hence the determinant will become 0.
So, x^{2} is a factor.
By adding row 1, row 2 and row 3, we get
4 - x + 4 + x + 4 + x = 0
12 + x = 0
x = -12
Hence -12 is the value which make the determinant zero. So the answers are 0, 0 and -12.
Question 2 :
Show that
Solution :
let us apply, x = y
Column 1 and 2 are identical. So the determinant will become zero.
Hence (x - y) is a factor. In the same way, we may show that (y - z) and (z - x) are factors.
Sum of exponents of leading diagonal = 3
A number of factors that we get so far = 3
Hence the required factor is a constant (k).
1(18 - 12) - 1(9 - 3) + 1(4 - 2) = k(-1)(-1)(2)
6 - 6 + 2 = 2k
k = 1
By applying the value of k, we get the given proof.
Question 3 :
In a triangle ABC, if
prove that triangle ABC is an isosceles triangle.
Solution :
By putting sin A = sin B, we get
That is, by putting sin A = sin B we see that, the given equation is satisfied.
Similarly by putting sin B = sin C and sin C = sin A, the given equation is satisfied.
Thus, we have A = B or B = C or C = A.
In all cases atleast two angles are equal. Thus the triangle is isosceles.
After having gone through the stuff given above, we hope that the students would have understood, "Factor Theorem Examples and Solutions".
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