Consider the polynomial function: f(x)=x^3-7x^2-4x+78

Question

(a) Find all the possible rational zeros of the function.

(b) Find all the zeros of the function

Answer 1

By Rational Zeros Theorem List of Possible Rational Zeros: ±1, ±2, ±3, ±6, 13, ±26, ±78.

Let check that x = - 3 is zero, using synthetic division:

-3 | 1 -7 -4 78

-3 30 -78

1 -10 26 |0

Now we set quotions x² - 10x + 26 = 0, x = 5 ± i

Zeros set:{-3, 5 + i, 5 - i}

Answer 2

p = all factors of 78

q = all factors of 1

p/q = 1, 2, 3, 6, 13, 26, 39, 78

Test each to see if it divides into the polynomial with no remainder.