Explanation:
If a polynomial function has roots at x=-5 and x=√3, it means that it can be factored as follows:
h(x) = a(x+5)(x-√3)
where 'a' is a constant coefficient that represents the leading term of the polynomial function.
To find the value of 'a' and express the polynomial function in standard form, we can multiply the factors and simplify as follows:
h(x) = a(x+5)(x-√3)
h(x) = a(x^2 - √3x + 5x - 5√3)
h(x) = a(x^2 + 2x√3 - 5√3)
To find the value of 'a', we can substitute one of the given roots into the equation and solve for 'a'. Let's choose x = -5:
h(-5) = a((-5)^2 + 2(-5)√3 - 5√3)
h(-5) = a(25 - 10√3 - 5√3)
Since x=-5 is a root, h(-5) = 0. Therefore:
0 = a(25 - 15√3)
a = 0 or a = (15√3)/25 = (3√3)/5
Since the coefficient 'a' cannot be zero, the polynomial function h(x) of lowest degree in standard form is:
h(x) = (3√3/5)(x^2 + 2x√3 - 5√3)
h(x) = (3√3/5)x^2 + (6/5)√3x - 3√3