solve equations

Question

1. f(x)=x^3-4x^2+4x-16 zeroes;2i 
2. f(x)= 2x^4+5x^3+5x^2+20x-12 zeroes;-2i 
3.h(x)= x^4-9x^3+21x^2+21x-130 zeroes; 3-2i?

Answer 1

1) The function f(x) = x³ - 4x² + 4x - 16

y = x³ - 4x² + 4x - 16

To find zeros of function substitute y = 0 in above equation.

 x³ - 4x² + 4x - 16 = 0

x²(x - 4) + 4(x - 4) = 0

(x - 4)(x² + 4) = 0

Factoring of x³ - 4x² + 4x - 16 is (x - 4)(x² + 4).

The above cubic polynomial have 3 zeroes.

2i is one of the root of the function.

From Complex conjugate root theorem, the imaginary roots are come in pairs.

So - 2i is must be root of the polynomial.

(x + 2i)(x - 2i) = x² - 2ix + 2ix - 4i²

= x² - 4(- 1) = x² + 4

To find remaining zero, just divide the (x - 4)(x² + 4) by (x² + 4).

 (x - 4)(x² + 4)/(x² + 4) = 0

x - 4 = 0

x = 4

Therefore, zeros of f(x) are at 4 and 2i, - 2i.

Answer 2

2) The function f(x) = 2x⁴ + 5x³ + 5x² + 20x - 12

If p/q is a rational zero, then p is a factor of 12 and q is a factor of 2.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 3, ± 4, ± 6 , ± 12 , ± 1/2 and ± 3/2.

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

p/q	2	5	5	20	-12
1	2	7	12	32	20
- 1	2	3	2	18	-30
2	2	9	23	66	120
-2	2	1	3	14	-40
-3	2	-1	8	-4	0

Since f(- 3) = 0, x = - 3 is a zero. The depressed polynomial is 2x³ - x² + 8x - 4 = 0.

Comments

Contd....

If p/q is a rational zero, then p is a factor of 4 and q is a factor of 2.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2 ,± 1/2 and ± 4.

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

p/q	2	- 1	8	- 4
1	2	1	9	5
2	2	3	14	24
1/2	2	0	8	0

Since f(1/2) = 0, x = 1/2 is a zero. The depressed polynomial is  2x² + 8.

Solve the equation 2x² + 8 = 0

x² + 4 = 0

x²  = 4

Apply square root on each side.

x = ± √(-4)

x = 2i and - 2i

Zeros of the f(x) = 2x⁴ + 5x³ + 5x² + 20x - 12 are at -3, 1/2 and ± 2i.

Answer 3

3) The function h(x) = x⁴ - 9x³ + 21x² + 21x - 130

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 130 and q is a factor of 1.

The possible values of p are  ± 1, ± 2, ± 5,± 10,± 13,± 26 and ± 130 .

The possible values for q are ± 1

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 5,± 10,± 13,± 26 and ± 130.

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

p/q	1	-9	21	21	-130
1	1	-8	13	34	-96
-1	1	-10	31	-10	-120
-2	1	-11	43	-65	0

Since f(-2) = 0, x = - 2 is a zero. The depressed polynomial is  x³ - 11x² + 43x - 65 = 0.

If p/q is a rational zero, then p is a factor of 65 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ±3, ±5, ±13 and ± 65.

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

Answer 4

Contd....

p/q	1	-11	43	-65
1	1	-10	33	-32
2	1	-9	25	-15
5	1	-6	13	0

Since f(5) = 0, x = 5 is a zero. The depressed polynomial is  x² - 6x + 13.

Since the depressed polynomial of this zero, x² - 6x + 13, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

x = 3 ± 2i

Zeros of the f(x) = x⁴ - 9x³ + 21x² + 21x - 130 are at -2, 5 and 3 ± 2i.