# f(x) = x^4 – x^3 + 7x^2 – 9x – 18

f(x) = x4 - x3 + 7x2 - 9x -18

x4 - x3 + 7x2 - 9x -18 = 0

Synthetic division theorem, possible roots are ±1, ±2, and ±3.

-1 | 1    -1    7     -9     -18

|  0    -1    2     -9      18

|_______________________

2 | 1     -2    9    -18     0

| 0      2    0      18

|_______________________

1      0     9      0

By checking , we find that -1 and 2 are a roots.

By the fundamental theorem of algebra.

The remaining root are those (x2 + 9) = 0

Subtract 9 from each side.

x2 = - 9

Apply square root each side.

x = √(-9) = √(9i2) = ± 3i                   [i2 = -1]

So the roots are  -1, 2, 3i and -3i.

Thank you!!! How would you know the possible roots? any rule to be applied?

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation anxn + an  1xn – 1 + ... + a1x + a0 = 0, then p is a factor of a0 and q is a factor if an.

The function f (x) = x4 - x3 + 7x2 - 9x - 18.

If p/q is a rational zero, then p is a factor of 18 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, ± 9.

The possible values for q are ± 1

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 9.

Make a table for the synthetic division and test possible real zeros.

 p/q 1 - 1 7 - 9 18 1 1 0 7 - 2 -16 - 1 1 - 2 9 - 18 0

Since f(- 1) = 0, x = – 1 is a zero. The depressed polynomial is  x3 - 2x2 + 9x - 18 = 0.

Contd.............

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 9.

Make a table for the synthetic division and test possible real zeros.

 p/q 1 - 2 9 - 18 1 1 - 1 8 - 10 - 1 1 - 3 12 - 30 2 1 0 9 0

Since f(2) = 0, x = 2 is a zero. The depressed polynomial is  x2 + 9 = 0.

The depressed polynomial is  x2 + 9 = 0.

x2 = - 9

x2 = (± 3i)2

x = - 3i and x = 3i.

The function has two real zeros at x = -1 and x = 2, two imaginary zeros at x = - 3i and

x = 3i.