Factor​ f(x) into linear factors given that k is a zero of​ f(x).

Question

f(x)=3x^3-5x^2-34x+24; k=4

f(x)= ?

Answer 1

To factor f(x) into linear factors given that k=4 is a zero of f(x), we can use synthetic division to find the quotient and the remaining factor.

Using synthetic division with 4 as the zero, we get:

4 | 3 -5 -34 24

| 12 28 -24

|-------------

3 7 -6 0

Therefore, the quotient is 3x^2+7x-6, and the remaining factor is (x-4). Thus, we can write:

f(x) = (x-4)(3x^2+7x-6)

To further factor 3x^2+7x-6, we can use the quadratic formula or factoring by grouping. Using factoring by grouping, we can write:

f(x) = (x-4)(3x^2+9x-2x-6)

= (x-4)(3x(x+3)-2(x+3))

= (x-4)(3x-2)(x+3)

Therefore, the complete factorization of f(x) is:

f(x) = (x-4)(3x-2)(x+3)

Answer 2

Use synthetic division to divde f(x) by x - 4.

Factor the suppres quadraticc polynomial.