The function which has real zero at x = –8 and x = 5 is x² + 3x - 40
Given the real zeroes at x = -8 and x = 5.
Factor theorem states that (x-r) is a factor of the polynomial function f(x) if and only if r is a root of the function f(x).
Since, we know that the root of the function i.e g(x) are -8 and 5 then the function has the following factor:
(x+8) = 0 and (x-5) =0
Zero product property states that if ab = 0 if and only if a =0 and b =0.
By zero product property,
(x+8)(x-5) = 0
Now, distribute each terms of the first polynomial to every term of the second polynomial we get;
x(x - 5) + 8(x - 5)
Now, when you multiply two terms together you must multiply the coefficient (numbers) and add the exponent.
x² + 8x - 5x - 40
Combine like terms;
x² + 3x - 40
therefore, the function has real zeros at x = -8 and x =5 is x² + 3x - 40
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