Write the simplest polynomial in factored form that has zeros at -2, 5, and 7.
To find the polynomial in factored form with the given zeros (-2, 5, and 7), we can use the fact that the factors of the polynomial will be (x - 0) for each zero.
First, let's write out the factor using each zero:
(x - (-2)) = (x + 2) (zero at -2)
(x - 5) (zero at 5)
(x - 7) (zero at 7)
To find the complete polynomial, we need to multiply these factors together.
Multiplying all the factors:
P(x) = (x + 2)(x - 5)(x - 7)
Expanding the expression:
P(x) = (x^2 - 5x - 7x + 35)(x + 2)
Simplifying further:
P(x) = (x^2 - 12x + 35)(x + 2)
Finally, let's multiply the remaining factors:
P(x) = x^3 + 2x^2 - 12x^2 - 24x + 35x + 70
Simplifying and combining like terms:
P(x) = x^3 - 10x^2 + 11x + 70
So, the simplest polynomial in factored form with zeros at -2, 5, and 7 is: P(x) = x^3 - 10x^2 + 11x + 70.