How would you write a polynomial function with rational coefficients so that P(x)=0 has the given roots?
The given roots are -2,-2,3,5
p(x) = (x+2)(x+2)(x-3)(x-5)
you can expand that if you want.
To write a polynomial function with rational coefficients using the given roots, we need to use the factors (x - root) for each root.
Given roots: -2, -2, 3, 5
To take into account the repeated root -2, we square the factor (x + 2) twice:
(x + 2)^2
The other two roots, 3 and 5, can be written as (x - 3) and (x - 5), respectively.
Combining all the factors, the polynomial function with rational coefficients is:
P(x) = (x + 2)^2(x - 3)(x - 5)
To write a polynomial function with rational coefficients based on the given roots, you can use the fact that a polynomial function can be expressed as the product of its linear factors.
In this case, the given roots are -2, -2, 3, and 5. Since -2 is repeated as a root, it means that (x + 2) is a factor of the polynomial function. Similarly, (x - 3) and (x - 5) are also factors, since 3 and 5 are given as distinct roots.
To find the polynomial function, you multiply these linear factors together:
P(x) = (x + 2)(x + 2)(x - 3)(x - 5)
Expanding this expression, you get:
P(x) = (x^2 + 4x + 4)(x - 3)(x - 5)
Multiplying further, you get:
P(x) = (x^2 + 4x + 4)(x^2 - 8x + 15)
Expanding again, you have:
P(x) = x^4 - 4x^3 - 7x^2 + 52x - 60
Therefore, the polynomial function with rational coefficients that has the roots -2, -2, 3, and 5 is P(x) = x^4 - 4x^3 - 7x^2 + 52x - 60.