12. Write a polynomial in factored form with zeros \( 5,6,7 \) and \( a=1 \)

To write a polynomial in factored form with zeros at \( 5, 6, \) and \( 7 \), you can use the fact that the polynomial can be expressed as \( (x - 5)(x - 6)(x - 7) \). Since \( a = 1 \), you don’t need to include any additional coefficients. So, the polynomial is: \[ f(x) = (x - 5)(x - 6)(x - 7) \] If you expand this, you’ll end up with a cubic polynomial, but in this case, we’re celebrating the beauty of the factored form! An interesting aspect of polynomials is how they relate to the shapes of their graphs. Since we have three distinct zeros, the graph of this polynomial will intersect the x-axis at the points \( 5, 6, \) and \( 7 \). Each zero corresponds to a "bounce" or "pass-through" at the x-axis, giving your polynomial a distinct shape and a chance for some creative interpretation if you're into sketching!