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A linear Factor of the Function, 3) F(x) = X4 - 583+318²-125 xt 150

User Kathlyn
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We are asked to find the linear factors of the polynomial:

f(x) = x^3 + 4 x^2 + 25 x + 100

So we start by trying to find a real zero for it using the rational zeros theorem.

We find that the rational number "-4) is one root of the polynomial, since when replacing it for "x" we get that:

f(-4) = 0

So we know that (x- -4) =(x + 4) is a binomial factor of the original polynomial.

Now, by using long division, we find the quadratic polynomial quotient:

f(x) = (x + 4) * (x^2 + 25)

So, now this quadratic factor (x^2 + 25) doesn't have real roots, but COMPLEX roots:

x^2 + 25 = 0

solving for x:

x^2 = - 25

Then the possible solutions are:

x = 5 i and x = -5i

which gives us the linear factors: ( x - 5 i ) and (x + 5 i)

Then the total factorization in linear terms is:

f(x) = (x + 4) (x - 5i) (x+ 5 i)

which agrees with option "C" in the provided list of possible solutions.

User Zwbetz
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