Find the complex zeros of the polynomial function. Write f in factored form.

Question

f(x)=x^4+10x^3-2x^2+90x-99

The complex zeros of f are

Use the complex zeros to write f in factored form F(x)=

Answer 1

Given f(x) = x⁴ +10x³ - 2x²  + 90x -99 

Rational Root Theorem, if a rational number in simplest form p /q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

If p /q is a rational zero, then p  is a factor of -99 and q  is a factor of 1.

The possible values of p  are  ± 1 , ± 3,   ± 9, ± 11 ,±99.

The possible values for q  are ± 1

By the Rational Roots Theorem, the only possible rational roots are, p /q  = ± 1 , ± 3,   ± 9, ± 11 , ±99

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

p /q	1	10	-2	90	-99
1	1	11	9	99	0

Since f (1) = 0,   x = 1 is a zero. The depressed polynomial is  x³ +11x² + 9x + 99 = 0.

 

Now   x³ +11x² + 9x + 99 = 0

If p/q is a rational zero, then p  is a factor of 99 and q  is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p /q  = ± 1 , ± 3,   ± 9, ± 11 , ±99

p /q	1	11	9	99
-1	1	10	-1	100
1	1	12	21	120
3	1	14	51	252
-3	1	8	-15	144
9	1	20	189	1800
-9	1	2	-9	180
-11	1	0	9	0

Since f (-11) =  0, x = -11 is a zero. The depressed polynomial is  x ²  + 9 = 0

x ²  + 9 = 0

x² = -9

x = ± 3i

Roots of the polynomial  at x = 1 and x = -11 and two imaginary roots at x = 3i and x = -3i.

The complex zeros of f(x) are 3i , -3i

 Factored form F(x)= (x-1)(x+11)(x+3i)(x-3i)