Question

Factor f(x) into linear factors given that k is a zero of f(x). f(x)=x* + 3x3 - 12x2 -52x - 48; k= -2 (multiplicity 2) In completely factored form, f(x)= ( (Factor completely.) x )

Answer 1

To factor the polynomial f(x) with k = -2 as a zero of multiplicity 2, include (x + 2)² in the factorization. Perform division to find the remaining quadratic, which will yield the complete factorization of f(x) into linear factors.

Step-by-step explanation:

Given that the polynomial f(x) = x^4 + 3x^3 - 12x^2 -52x - 48 has a zero of k = -2 with multiplicity 2, we can start factoring the polynomial by including (x + 2)² as part of the factorization.

Now we need to divide f(x) by (x + 2)² to find the other factors. Once we have completed the long division or synthetic division, we will have the remaining quadratic, which will factor or can be solved by the quadratic formula to give us the remaining linear factors of f(x).

If we find that the resulting quadratic doesn't factor easily, we apply the quadratic formula, which is x = (-b ± √(b² - 4ac))/(2a), to find the rest of the zeros. This process will yield the completely factored form of the given polynomial, which will be the product of linear factors including the given factor with its multiplicity.