Question

question from polynomial

please solve it​

Answer 1

Step-by-step explanation:

Let f(x) = 2x⁴-13x³+19x²+7x-3

Let α= (2+√3) and β= (2-√3).

Then, (α+β) = 4 and αβ = (4-3) = 1.

So, the quadratic polynomial whose roots are α & β is given by

x²-(α+β)x+αβ = (x²-4x+1).

∴ (x²-4x+1) is a factor of f(x).

On dividing f(x) by (x²-4x+1), we get

x²-4x+1 ) 2x⁴-13x³+19x²+7x-3 ( 2x²-5x-3

2x⁴ -8x³+ 2x²

- + -

-5x³ +17x²+7x-3

-5x³+20x²-5x

+ - +

-3x²+12x-3

-3x²+12x-3

+ - +

×

∴ f(x) = (x²-4x+1) (2x²-5x-3)

∴ the other two zeros of f(x) are given by (2x²-5x-3) = 0.

Now, 2x²-5x-3 = 0 ⇒2x²+x-6x-3 = 0

⇒x(2x+1)-3(2x+1) = 0

⇒(2x+1) (x-3) = 0

⇒2x+1 = 0 or x-3 = 0

⇒x = -1/2 or x = 3

Hence, the other two zeros of f(x) are -1/2 and 3.