How to Find all the real roots of

Question

3X^4+X^3-9X^2+3X+2?

Answer 1

The polynomial function is 3x ⁴ + x ³ - 9x ² + 3x + 2 = 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 2 and q is a factor of 3.

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

The possible values for q are ± 1, ± 3

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

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

p/q	3	1	-9	3	2
-1	3	-2	-7	10	-8
1	3	4	-5	-2	0

Since f(1)   =   0,  x   =  1 is a zero. The depressed polynomial is 3x ³ + 4x ² – 5x - 2 = 0.

 

• If p/q is a rational zero, then p is a factor of 2 and q is a factor of 3.

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

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

p/q	3	4	-5	-2
-1	3	1	-6	4
1	3	7	2	0

Since f(1)  =  0,   x = 1 is a zero. The depressed polynomial is  3x ² + 7x + 2 = 0

• Since the depressed polynomial of this zero, 3x ² + 7x + 2, is quadratic, use the  Factorisation to find the roots of the  equation 3x ² + 7x + 2 = 0

3x ² + 6x + x + 2 = 0

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

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

Apply zero product property.

x + 2 = 0 and 3x + 1 = 0

x = -2 and 3x = -1

x = -2 and x = -1/3

3x ⁴ + x ³ - 9x ² + 3x + 2 have 3 real roots at x = -2 and x = -1/3 and x = 1.