what are the real roots and what are the imaginary roots: f(x)=10x^5-15x^4+12x^3-18x^2+2x-3

Answer 1

By rational Zeros Theorem: ±1, ±3 factors of a₀ = 3; ±1, ±2, ±5, ±10 factors of a₅ = 10.

Then list of possible rational zeros: ±1, ±3, ±1/2, ±3/2, ±1/5, ±3/5, ±1/10, ±3/10.

Then we can use synthetic division to find zeros if any from this list. I will use graph of left side polynomial with TI-84. Look like 3/2 only real solution. Let check using synthetic division:

3/2 | 10 -15 12 -18 2 -3

15 0 18 0 3

10 0 12 0 2 0

So we get that x = 3/2 is zero.and coetion is 10x⁴ + 12x² + 2 = 0 or 5x⁴ + 6x² + 1 = 0

WE can factor now left side:(5x² + 1)(x² + 1) = 0, x² + 1 = 0, x² = - 1, x = ±i

or 5x² + 1 = 0, x² = - 1/5, x = ±i/√5.

This is zeros set of this equation: {3/2, i, - i, i/√5, - i/√5}