Create a polynomial \( p \) that has the desired characteristics. You may leave the polynomial in factored form. - The \( x \) intercepts of \( p \) are \( (6,0),(9,0),(-4,0) \) and \( (-5,0) \). - The leading term of \( p \) is \( 125 x^{4} \). \( p(x)= \)

To construct the polynomial \( p(x) \) with the given characteristics, follow these steps: 1. **Identify the Roots:** The \( x \)-intercepts \( (6,0) \), \( (9,0) \), \( (-4,0) \), and \( (-5,0) \) indicate that the roots of the polynomial are \( x = 6 \), \( x = 9 \), \( x = -4 \), and \( x = -5 \). 2. **Form the Factored Polynomial:** For each root \( r \), the corresponding factor is \( (x - r) \). Therefore, the factored form of the polynomial is: \[ p(x) = a(x - 6)(x - 9)(x + 4)(x + 5) \] where \( a \) is the leading coefficient. 3. **Determine the Leading Coefficient:** The leading term of \( p(x) \) is given as \( 125x^4 \). When multiplying out the factors, the coefficient of \( x^4 \) is \( a \). Therefore, to achieve a leading term of \( 125x^4 \), set \( a = 125 \). 4. **Final Polynomial:** Substituting \( a = 125 \) into the factored form gives: \[ p(x) = 125(x - 6)(x - 9)(x + 4)(x + 5) \] **Final Answer:** \[ p(x) = 125\, (\,x - 6\,)(\,x - 9\,)(\,x + 4\,)(\,x + 5\,) \]