2x^4 - 5x^3 - 11x^2 + 20x + 12; -2, 3 find roots of polynomials.

Question

please  help!

Answer 1

Solve by using synthetic division.

Given roots of the equation .

The quotient is .

Solve for roots

.

The roots of the polynomial are

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.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nx ⁿ + 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 2x ⁴ - 5x ³ - 11x ²+ 20x + 12.

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

The possible values of p are   ± 1,   ± 3, and ± 4.

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

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

Make a table for the synthetic division and test, wheather - 2 is a possible real zero or not.

p/q	2	- 5	- 11	20	12
- 2	2	- 9	7	6	0

Since, f(- 2) = 0, x = - 2 is a zero. The depressed polynomial is  2x ³ - 9x ² + 7x + 6 = 0.

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 2.

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

Make a table for the synthetic division and test, wheather 3 is a possible real zero or not.

p/q	2	- 9	7	6
3	2	- 3	- 2	0

Since, f(3) = 0, x = 3 is a zero. The depressed polynomial is  2x ² - 3x - 2 = 0.

By factor by grouping.

2x ² - 4x + x - 2 = 0

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

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

Apply zero product property.

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

⇒ x = 2 and x = - 1/2.

Therefore, the roots of the polynomial are x = - 2, x = 3, x = 2, and x = - 1/2.