Find a polynomial function of degree 3 with the given numbers as zeros.
-5, sqrt3, -sqrt3
(x+5)(x+sqrt3)(x-sqrt3) = 0
Multiplying out the left side gives you:
(x+5)*(x^2-3) = 0
x^3 +5x^2-3x-15 = 0
To find a polynomial function of degree 3 with the given zeros, we can use the fact that if r is a zero of a polynomial function, then (x - r) is a factor of the polynomial.
Let's break down the given zeros:
1. -5 is a zero, so (x - (-5)) = (x + 5) is a factor of the polynomial.
2. √3 is a zero, so (x - √3) is a factor of the polynomial. However, we need to consider both the positive and negative square root of 3 (i.e., (x - √3) and (x + √3)).
3. -√3 is a zero, so (x - (-√3)) = (x + √3) is a factor of the polynomial.
To find the polynomial, we multiply all the factors together. Thus, the polynomial function can be written as:
P(x) = (x + 5)(x - √3)(x + √3)
Simplifying the above expression, we get:
P(x) = (x + 5)(x^2 - (√3)^2)
Expanding further, we have:
P(x) = (x + 5)(x^2 - 3)
Multiplying the terms, we get:
P(x) = x^3 - 3x + 5x^2 - 15
Finally, combining like terms, we have:
P(x) = x^3 + 5x^2 - 3x - 15
Therefore, the polynomial function of degree 3 with the given zeros (-5, √3, -√3) is P(x) = x^3 + 5x^2 - 3x - 15.