Use the Rational Zeros Theorem to find all the zeros of the

Use the Rational Zeros Theorem to find all the zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = 5x^4 - 8x^3 + 13x^2 - 16x + 6 2, 3/5; f(x) = (x - 2) (5x - 3) (x^2 + 1) -2, -1, 1, -3/5; f(x) = (x - 1) (5x + 3) (x + 1) (x + 2) 1, 3/5; f(x) = (x - 1) (5x - 3) (x^2 + 2) -2, -1, 1, 3/5; f(x) = (x - 1) (5x - 3) (x + 1) (x + 2)

Solution

p = 6(last number)

Factors of p : 1,2,3,6

q = 5(leading coefficient)

So, factors of q : 1,5

Now, acc to rational zero theorem
possible rational roots = plus or minus of
\"factor of p\" / \"factor of q\"

So, this makes it :

Possible rational zeros are :
1,2,3,6,-1,-2,-3,-6,1/5,2/5,3/5,6/5,-1/5,-2/5,-3/5 and -6/5

Now, we plug in these rational zeros and check on the function...

For ex
Plug in x = 1
5 - 8 + 13 - 16 + 6 comes out 0
So, 1 is a zero and its corresponding factor is (x-1)

Similarly, 3/5 works

None of the others work....

So, we have factors (x-1)(5x-3)

Now, when we synthetically divide this out,
we get x^2 + 2 as the quotient

So, factored form is :
(x - 1)(5x - 3)(x^2 + 2)

Option C is the answer