Final answer:
To find the factors of f(x) given that x = 4 is a zero, we need to use polynomial division. Divide the polynomial by (x - 4) and repeat the process until there is no remainder. The divisors used in each step will be the factors of f(x).
Step-by-step explanation:
To find the factors of f(x) given that x = 4 is a zero, we need to divide the polynomial f(x) = x^3 - 7x^2 + 2x + 40 by the factor (x - 4) using polynomial division.
1. Arrange the polynomial in descending order of powers of x: f(x) = x^3 - 7x^2 + 2x + 40.
2. Divide the highest power term by the factor: (x^3)/(x-4) = x^2.
3. Multiply the factor by the quotient: (x-4)(x^2) = x^3 - 4x^2.
4. Subtract the result from the original polynomial: (x^3 - 7x^2 + 2x + 40) - (x^3 - 4x^2) = -3x^2 + 2x + 40.
5. Repeat steps 2-4 with the new polynomial obtained in step 4 until there is no remainder.
By the end of the division, you should have the quotient and the remainder. The factors of f(x) are the divisors used in each step of the division.