Precalculus problems HELP?

Question

Find all real solutions of the polynomial equation.
3y4 + 8y3 − 30y2 + 24y − 5 = 0

Find the rational zeros of the function.
C(x) = 2x3 + 3x2 − 1

Answer 1

The polynomial 3y⁴  + 8y³  - 30y²  + 24y - 5 = 0

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 5 and q is a factor of 3.

The possible values of p are   ± 1 and  ± 5.

The possible values for q are ± 1 and ± 3.

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

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

p/q	3	8	- 30	24	- 5
- 1	3	5	- 35	59	- 64
1	3	11	- 19	5	0

Since f(1) = 0,  y  =  1 is a zero. The depressed polynomial is  3y³ + 11y² - 19y + 5 = 0.

Comments

If p/q is a rational zero, then p is a factor of 5 and q is a factor of 3.

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

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

p/q	3	11	- 19	5
- 1	3	8	- 27	32
1	3	14	- 5	0

Since f(1) = 0,  y = 1 is a zero. The depressed polynomial is 3y² + 14y - 5 = 0

3y² + 15y + y - 5 = 0

3y(y + 5) - 1(y + 5) = 0

(y + 5)(3y - 1) = 0

y + 5 = 0 and 3y - 1 = 0

y = - 5 and y = 1/3

Real zeros of polynomial are y = 1 , 1 , 1/3 and - 5.

Answer 2

The polynomial function c(x) = 2x³ + 3x² - 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.

2x³ + 3x² - 1 = 0

If p/q is a rational zero, then p is a factor of 1 and q is a factor of 2.

The possible values of p are   ± 1.

The possible values for q are ± 1 and ± 2.

So, p/q = ± 1,   ±1/2.

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

p/q	2	3	0	- 1
1	2	5	5	4
- 1	2	1	- 1	0

Since c(- 1) = 0, x = - 1 is a zero. The depressed polynomial is   2x² + x - 1 = 0

Since the depressed polynomial of this zero, 2x² + x - 1, is quadratic, use Factorization to find the roots of the related quadratic equation.

2x² + 2x - x - 1 = 0

2x(x + 1) - 1(x + 1) = 0

(x + 1)(2x - 1) = 0

x + 1 = 0 and 2x - 1 = 0

x = - 1 and x = 1/2

Rational zeros of polynomial are at x = -1, - 1, 1/2.