Find all zeros (real or complex) of f(x) = x^5-4x^4+4x^3+10x^2-13x-14

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Answer 1

Given polynomial f(x) = x^5-4x^4+4x^3+10x^2-13x-14

By synthatic division

x^5-4x^4+4x^3+10x^2-13x-14 = (x+1)(x+1)(x-2)(x^2-4x+7)

Now Factorize (x^2-4x+7).

Compare it quadratic form ax^2+bx+c

a = 1, b = -4, c = 7

Roots are x = [-b±√b^2-4ac]/2a

x = [4±√(16-4*1*7)]/2*1

x = [4±√-12]/2

x = [4±√-3*4]/2

x = [ 4±2√-3]/2

x = 2[2±√3i]/2

x = 2±√3i

Zeros of given polynomial is x = -1,-1,2,2+√3i,2-√3i.

Answer 2

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.

The function is F (x) = x⁵ - 4x⁴ + 4x³ + 10x² - 13x - 14.

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 14 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, ± 7.

The possible values for q are ± 1

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ±7.

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

p/q	1	- 4	4	10	- 13	- 14
1	1	- 3	1	11	- 2	- 16
2	1	- 2	0	10	7	0

Since f(2)  =  0,  x  =  2 is a zero. The depressed polynomial is  x⁴ - 2x³ + 10x + 7 = 0.

Answer 3

Contd.....

If p/q is a rational zero, then p is a factor of  7 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 7.

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

p/q	1	- 2	0	10	7
1	1	- 1	- 1	9	16
-1	1	- 3	3	7	0

Since f(-1) = 0, x = - 1 is a zero.The depressed polynomial is  x³- 3x² + 3x + 7 = 0.

If p/q is a rational zero, then p is a factor of  7 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 7.

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

p/q	1	- 3	3	7
1	1	- 2	1	8
- 1	1	- 4	7	0

Since f(-1) = 0, x = - 1 is a zero.The depressed polynomial is  x² - 4x + 7 = 0.

Since the depressed polynomial of this zero, x² - 4x + 7 = 0, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

Substitute a = 1, b = - 4, and c = 7.
 

Substitute i ² = - 1.

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