Final answer:
To factor f(x) into linear factors, divide f(x) by (x - k), where k is the zero. In this case, f(x) = x^3 - 48x - 128 and k = -4.
Therefore, the linear factors are (x + 4)(x^2 - 4x - 32). So, the correct option is b) x - 4.
Step-by-step explanation:
To factor f(x) into linear factors, given that k is a zero of f(x), we can use synthetic division. We divide f(x) by (x - k), where k is the zero. In this case, k = -4. So, we divide f(x) by (x + 4). The quotient will be the linear factors.
Using synthetic division, we have:
-4 | 1 0 -48 -128
-4 16 128
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1 -4 -32 0The subject at hand involves factoring the polynomial f(x) = x^3 - 48x - 128 into linear factors, and we are told that k = -4 is a zero of f(x). To find the linear factors of f(x), we can use the given zero to perform polynomial division or apply synthetic division to factor out (x + 4).Once we factor out (x + 4), we'll be left with a quadratic equation that we can solve using the quadratic formula or further factoring if possible, to find the remaining zeros. Each zero corresponds to a linear factor of the polynomial. The correct choice from the given options is a) x + 4 since k = -4 is a root of the polynomial.
The quotient is x^2 - 4x - 32. Therefore, the linear factors are (x + 4)(x^2 - 4x - 32). So, the correct option is b) x - 4.