Application of Factor Theorem

We will discuss here about the application of Factor Theorem. 

1. Find the roots of the equation 2x^2 – 7x + 6 = 0. Hence factorize 2x^2 – 7x + 6.

Solution:

Here, the equation is 2x^2 – 7x + 6 = 0

Or, 2x^2 – 4x – 3x + 6 = 0

2x(x – 2) – 3(x – 2) = 0

(x – 2)(2x – 3) = 0

X – 2 =0 or 2x – 3 = 0

X = 2 or x = \(\frac{3}{2}\)

Therefore, 2x^2 – 7x + 6 = 2(x – 2)(x - \(\frac{3}{2}\)) = (x – 2)(2x – 3)


2. Find the quadratic equation whose roots are 1 + √3 and 1 - √3.

Solution:

We know that the quadratic equation whose roots are α and β, is

(x – α)(x – β) = 0

Therefore, the required equation is {x - (1 + √3)}{x - (1 - √3)} = 0

Or x^2 - {1 - √3 + 1 + √3}x + (1 + √3)( 1 - √3) = 0

Or, x^2 – 2x + (1 – 3) = 0

X^2 – 2x – 2 = 0.



3. Find the cubic equation whose roots are 2, √3 and -√3.

Solution:

We know that the quadratic equation whose roots are α, β and γ, is

(x – α)(x – β)(x - γ) = 0

Therefore, the required equation is (x – 2)(x - √3){x – (-√3)} = 0

(x - 2)(x - √3)(x + √3) = 0

(x - 2)(x^2 - 3) = 0

X^3 – 2x^2 – 3x + 6 = 0.

 

Or x^2 - {1 - √3 + 1 + √3}x + (1 + √3)( 1 - √3) = 0

Or, x^2 – 2x + (1 – 3) = 0

X^2 – 2x – 2 = 0.



4. Factorize x^2 -3x - 9

Solution:

The corresponding equation is x^2 - 3x - 9= 0

Now we apply the quadratic formula

x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

   = \(\frac{-(-3) \pm \sqrt{(-3)^{2} - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1}\)

   = \(\frac{3 \pm \sqrt{9 + 36}}{2}\)

   = \(\frac{3 \pm \sqrt{45}}{2}\)

   = \(\frac{3 \pm 3\sqrt{5}}{2}\)

Therefore, x^2 – 3x – 9 = (x - \(\frac{3 + 3\sqrt{5}}{2}\))(x - \(\frac{3 - 3\sqrt{5}}{2}\))

 












10th Grade Math

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