ind a degree 4 polynomial having zeros \( -6,-1,2 \) and 7 and the coefficient of \( x^{4} \) equal 1 . The polynomial is

To find a degree 4 polynomial with the given zeros \( -6, -1, 2, \) and \( 7 \), we can use the fact that if \( r \) is a zero of the polynomial, then \( (x - r) \) is a factor of the polynomial. Therefore, the polynomial can be expressed as: \[ P(x) = (x + 6)(x + 1)(x - 2)(x - 7) \] Since the coefficient of \( x^4 \) is 1, we can directly expand this product: 1. First, pair the factors: \[ (x + 6)(x + 1) = x^2 + 7x + 6 \] \[ (x - 2)(x - 7) = x^2 - 9x + 14 \] 2. Now, multiply these two quadratic polynomials: \[ P(x) = (x^2 + 7x + 6)(x^2 - 9x + 14) \] 3. Using the distributive property (FOIL), we get: \[ = x^4 - 2x^3 - 55x^2 + 108x + 84 \] Thus, the polynomial you are looking for is: \[ P(x) = x^4 - 2x^3 - 55x^2 + 108x + 84 \]