Find a polynomial of degree 4 that has a integer coefficients and zeros 1,-1,2 and 1/2 (one half)?
x ^ 4 + a x ^ 3 + b x ^ 2 + c x + d =
( x - x1 ) ( x - x2 ) ( x - x3 ) ( x - x4 ) =
( x - 1 ) [ x - ( - 1 ) ] [ x - 2 ) ] ( x - 1 / 2 )
( x - 1 ) ( x + 1 ) ( x - 2 ) ( x - 1 / 2 ) =
x ^ 4 - 5 x ^ 3 / 2 + 5 x / 2 - 1
To find a polynomial of degree 4 with integer coefficients and zeros at 1, -1, 2, and 1/2, we can use the fact that a polynomial with these given zeros can be written in factored form as the product of its linear factors.
Since 1, -1, 2, and 1/2 are the zeros, the linear factors can be written as:
(x - 1)(x + 1)(x - 2)(x - 1/2)
To obtain integer coefficients, we need to simplify the expression. By combining similar terms, we can expand and multiply the factors:
(x - 1)(x + 1)(x - 2)(x - 1/2) = (x^2 - 1)(x^2 - 5/2x + 1)
Next, we will expand the expression further:
(x^2 - 1)(x^2 - 5/2x + 1) = x^4 - (5/2)x^3 + x^2 - x^2 + (5/2)x - 1
Now we can simplify the expression to obtain the polynomial:
x^4 - (5/2)x^3 + (5/2)x - 1
Thus, the polynomial of degree 4 with integer coefficients and zeros at 1, -1, 2, and 1/2 is x^4 - (5/2)x^3 + (5/2)x - 1.