Write a polynomial equation with these roots.
a)-1,2+�ã3,2-�ã 3
b)2,5,1
To write a polynomial equation with given roots, we can use the fact that if r is a root of a polynomial equation, then (x-r) is a factor of the polynomial.
a) The roots are -1, 2+√3, and 2-√3.
To form a polynomial equation, we can multiply the factors (x-(-1)), (x-(2+√3)), and (x-(2-√3)), and simplify.
(x-(-1))(x-(2+√3))(x-(2-√3))
Expanding this expression, we get:
(x+1)(x-2-√3)(x-2+√3)
Now we can simplify further by multiplying the expressions:
(x+1)(x^2 - (2-√3)x - (2+√3)x + (2-√3)(2+√3))
Simplifying,
(x+1)(x^2 - 2x - √3x - 2x + 4 - √3 + 2√3 -3)
(x+1)(x^2 - 4x + 1 - √3)
Finally, expanding this, we have our polynomial equation:
x^3 - 4x^2 + (1 - √3)x^2 + x - 4 + √3
Therefore, the polynomial equation with roots -1, 2+√3, and 2-√3 is:
x^3 - 4x^2 + (1 - √3)x^2 + x - 4 + √3 = 0
b) The roots are 2, 5, and 1.
Following the same process, we'll multiply the factors (x-2), (x-5), and (x-1) to form a polynomial equation.
(x-2)(x-5)(x-1)
Expanding this expression:
(x^2 - 5x - 2x + 10)(x-1)
(x^2 - 7x + 10)(x - 1)
Now, we can multiply to obtain the final form of the polynomial equation:
x^3 - 7x^2 + 10x - x^2 + 7x - 10
x^3 - 8x^2 + 17x - 10
Therefore, the polynomial equation with roots 2, 5, and 1 is:
x^3 - 8x^2 + 17x - 10 = 0