obtain all other zeroes of

Question

3x^4+6x^3-2x^2-10x-5 if two o its zeroes are

Answer 1

The polynomial function 3x⁴ + 6x³ - 2x² – 10x - 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	6	-2	-10	-5
1	3	9	7	-3	-8
-1	3	3	-5	-5	0

Since f(-1)   =   0,   x  =  –1 is a zero. The depressed polynomial is 3x³ + 3x² – 5x - 5 = 0.

 

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,  ± 1/3, ± 5 and ± 5/3.

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

p/q	3	3	-5	-5
1	3	6	-1	-6
-1	3	0	-5	0

Since f(- 1)  =   0,   x = - 1 is a zero. The depressed polynomial is  3x² – 5 = 0

Solve the equation 3x² – 5 = 0

3x² = 5

x²  = 5/3

x = ± √(5/3)

Zeros are x = -1 ,-1,√(5/3) and - √(5/3).