How do you factor all zeros of y=6x^4+35x^3+35x^2-55x-21

Question

I need to factor and find all real zeros.

Answer 1

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 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.

The polynomial is P (x ) = y = 6x⁴ + 35x³ + 35x²- 55x - 21.

If p/q is a rational zero, then p is a factor of 21 and q is a factor of 6.

The possible values of p are   ± 1, ± 3, ± 7.

The possible values for q are ± 1, ± 2, ± 3.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 3, ± 7, ± 1/2, ± 3/2, ± 1/3, ± 7/3, ±7/2.

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

p/q	6	35	35	- 55	- 21
1	6	41	76	21	0

Since f(1) = 0, x = 1 is a zero. The depressed polynomial is  6x³ + 41x² + 76x + 21 = 0.

Answer 2

Contd....

If p/q is a rational zero, then p is a factor of 21 and q is a factor of 6.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 3, ± 7, ± 1/2, ± 3/2, ± 1/3, ± 7/3, ±7/2.

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

p/q	6	41	76	21
1	6	47	113	134
- 1	6	35	41	- 20
-2	6	29	18	3
-3	6	23	7	0

Since f(-3) = 0, x = -3 is a zero. The depressed polynomial is  6x² + 23x + 7 = 0.

Since the depressed polynomial of this zero, 6x² + 23x + 7 = 0, is quadratic, use the factor by grouping method to find the roots of the related quadratic equation 6x² + 23x + 7 = 0.

6x² + 23x + 7 = 0

Factor by grouping.

6x² + 2x + 21x + 7 = 0

2x(3x + 1) + 7(3x + 1) = 0

Factor : (3x + 1)(2x + 7) = 0

Apply zero product property.

(3x + 1) = 0 (or) (2x + 7) = 0

x = - 1/3 , - 7/2

The polynomial has all real zeros at x = 1, - 3, - 1/3, and - 7/2.