To find a quadratic polynomial given its zeroes, we can use the fact that if the zeroes of a polynomial are α and β, the polynomial can be expressed in the form:
f(x) = k(x – α)(x – β)
Here, α and β are the zeroes, and k is a constant. Since we are looking for a polynomial with the zeroes 3 and 4, we can substitute these values into the equation:
f(x) = k(x – 3)(x – 4)
If we want the simplest polynomial, we can take k = 1:
f(x) = (x – 3)(x – 4)
Now, let’s expand this expression:
f(x) = x² - 4x - 3x + 12
= x² - 7x + 12Thus, the quadratic polynomial whose zeroes are 3 and 4 is:
f(x) = x² – 7x + 12